Do the Catholic Church ex cathedra pronouncements about necessity of Catholicism to be saved still apply? I think the following was spoken Ex Cathedra:


  “The most Holy Roman Church firmly believes, professes and preaches
  that none of those existing outside the Catholic Church, not only
  pagans, but also Jews and heretics and schismatics, can have a share
  in life eternal; but that they will go into the eternal fire which was
  prepared for the devil and his angels, unless before death they are
  joined with Her; and that so important is the unity of this
  ecclesiastical body that only those remaining within this unity can
  profit by the sacraments of the Church unto salvation, and they alone
  can receive an eternal recompense for their fasts, their almsgivings,
  their other works of Christian piety and the duties of a Christian
  soldier. No one, let his almsgiving be as great as it may, no one,
  even if he pour out his blood for the Name of Christ, can be saved,
  unless he remain within the bosom and the unity of the Catholic
  Church.” (Council of Florence--Pope Eugene IV, the Bull Cantate
  Domino, 1441.)


So as far as 600 years ago, this was considered an infallible statement by Catholics, correct?

Here is another one from 700 years ago:


  “With Faith urging us we are forced to believe and to hold the one,
  holy, Catholic Church and that, apostolic, and we firmly believe and
  simply confess this Church outside of which there is no salvation nor
  remission of sin… Furthermore, we declare, say, define, and proclaim
  to every human creature that they by absolute necessity for salvation
  are entirely subject to the Roman Pontiff.” (Pope Boniface VIII, Unam
  Sanctam, Nov. 18, 1302)


Why then today do Catholics seem to change their mind about it?

There is indeed confusion about what Pope Francis meant about atheists. 

This answer on Stack Exchange states that there are other ways outside of the sacraments and union with the Catholic Church to get to heaven.

To be fair, it does contradict paragraphs 1257-1261 of the Catholic Catechism. 

So how does the Catholic Church regard these Ex Cathedra pronouncements which were clearly aimed at people trying to break away? Now that we have Protestantism, they seem to have mellowed a bit. But I would like to get a more in depth answer that explains


How Ex Cathedra statements seem to be disregarded
How Catholics actually squared these statements in the first place with the teachings in the Cathechism that seem to teach the exact opposite
When a Pope is Infallible
The First Vatican Council's Pastor Æternus said, under Pope Pius IX's authority, regarding papal infallibility:

…we teach and define that it is a dogma divinely revealed: that the
Roman Pontiff, when he speaks ex cathedra, that is, when in
discharge of the office of pastor and doctor of all Christians, by
virtue of his supreme Apostolic authority, he defines a doctrine
regarding faith or morals to be held by the universal Church, by the
divine assistance promised to him in blessed Peter, is possessed of
that infallibility with which the divine Redeemer willed that his
Church should be endowed for defining doctrine regarding faith or
morals; and that therefore such definitions of the Roman Pontiff are
irreformable of themselves, and not from the consent of the Church.

Thus, the following critera must hold for a pope's statement to be ex cathedra:
He must:

speak for the whole Church,
invoke all his authority,
intend to definitively define dogma, and
speak regarding the faith and/or morals.


Particular Examples
For example, Pope Eugene IV did meet all these criteria in Cantate Domino (1442):

speaks for the whole Church: &quot;The most Holy Roman Church firmly believes, professes and preaches…&quot;
invokes all his authority: The document is an authoritative papal bull.
intend to definitively define dogma: In this case, he is not defining anything new; he's only relaying what has been said before him (like in Unam Sanctam of 1302) regarding the necessity of the Church for salvation.
speak regarding the faith and/or morals: Yes. In this case, he speaks of both.

A similar analysis can be made for Pope Boniface VIII's Unam Sanctam (1302):

speaks for the whole Church: Bulls are addressed to the whole Church.
invokes all his authority: The document is an authoritative papal bull.
intend to definitively define dogma: &quot;…we declare, we proclaim, we define…&quot;
speak regarding the faith and/or morals: Yes. He speaks about what is necessary for salvation, which pertains to the faith.

Francis's statements do not meet these criteria.

speaks for the whole Church: This is unclear. Was he giving the homily only for those present in the Wednesday audience, or the whole Church? Certainly, modern media is able to make everything coming out of a pope's mouth sound like he's speaking for the whole Church.
invokes all his authority: Francis nowhere uses solemn language like, for example, Pope Pius XII used to declare the dogma of the Assumption of Mary into heaven in the Apostolic Constitution Munificentissimus Deus (1950):


For which reason, after we have poured forth prayers of supplication
again and again to God, and have invoked the light of the Spirit of
Truth, for the glory of Almighty God who has lavished his special
affection upon the Virgin Mary, for the honor of her Son, the immortal
King of the Ages and the Victor over sin and death, for the increase
of the glory of that same august Mother, and for the joy and
exultation of the entire Church; by the authority of our Lord Jesus
Christ, of the Blessed Apostles Peter and Paul, and by our own
authority, we pronounce, declare, and define it to be a divinely
revealed dogma:
that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul
into heavenly glory.
3. intend to definitively define dogma: Homilies have never been used to define dogma, nor did he express his intent to define dogma.
4. speak regarding the faith and/or morals: Yes, he was speaking about redemption.

Also, John Paul II's Catechism of the Catholic Church is fallible because it contradicts previous Church teaching on some points. See this for an in-depth analysis.

Whether a Pope Making Erroneous or Heretical Statements is Compatible with the Church's Sanctity
The 4 marks of the Church are:

unity
sanctity
Catholic (universality)
apostolic

The first mark, unity, does not just mean a unity of the members of the Church today. It means there must be a continuity in the Church's teachings and dogma from the time Christ founded the Church until today.
The second mark of the Church, its sanctity, means the Church cannot teach error or lead one into error.
What about when a bishop does teach error or heresy? For example, Pope Honorius I is thought to have taught the monothelite heresy (that Christ only has one will) in a private letter (see this for the controversy), and St. Frances de Sales, in his The Catholic Controversy, considered the possibility that he was a formal heretic and, since heretics are outside the Church, thus also an anti-pope. It's worth quoting the beginning of The Catholic Controversy (pp. 305 f.), the chapter on &quot;how ministers have violated their authority&quot;, which teaching influenced the First Vatican Council's definition on papal authority quoted above:

Under the ancient law the High Priest did not wear the Rational except
when he was vested in the pontifical robes and was entering before the
Lord. Thus we do not say that the Pope cannot err in his private
opinions, as did John XXII.; or be altogether a heretic as perhaps
Honorius was. Now when he is explicitly a heretic, he falls ipso
facto from his dignity and out of the Church, and the Church must
either deprive him, or, as some say, declare him deprived, of his
Apostolic See, and must say as S. Peter did : Let another take his
bishopric. [Acts 1] When he errs in his private opinion he must be
instructed, advised, convinced; as happened with John XXII, who was so
far from dying obstinate or from determining anything during his life
concerning his opinion, that he died whilst he was making the
examination which is necessary for determining in a matter of faith,
as his successor [Pope Benedict XII] declared in the Extravagantes
which begins Benedictus
Deus. But when
he is clothed with the pontifical garments, I mean when he teaches the
whole Church as shepherd, in general matters of faith and morals, then
there is nothing but doctrine and truth. And in fact everything a king
says is not a law or an edict, but that only which a king says as king
and as a legislator. So everything the Pope says is not canon law or
of legal obligation; he must mean to define and to lay down the law
for the sheep, and he must keep the due order and form.

Also, the following passage from St. Vincent Lerins's The Commonitory is worth quoting in full, too; it gives the rules for how to deal with a bishop or priest teaching error or heresy: stick to tradition.

Also in the Catholic Church itself we take great care that we hold
that which has been believed everywhere, always, by all. For that
is truly and properly Catholic, as the very force and meaning of the
word shows, which comprehends everything almost universally. And we
shall observe this rule if we follow universality, antiquity, consent.
We shall follow universality if we confess that one Faith to be true
which the whole Church throughout the world confesses; antiquity if
we in no wise depart from those interpretations which it is plain that
our ancestors and fathers proclaimed; consent if in antiquity itself
we eagerly follow the definitions and beliefs of all, or certainly
nearly all, priests and doctors alike.
What, then, will the Catholic Christian do if any part of the Church
has cut itself off from the communion of the Universal Faith? What
surely but prefer the soundness of the whole body to a pestilent and
corrupt member?
What if some novel contagion seek to infect the whole Church, and not merely a small portion of it? Then he will take care to cling to
antiquity, which cannot now be led astray by any novel deceit.
What if in antiquity itself error be detected on the part of two or
three men, or perhaps of a city, or even of a province? Then he will
look to it that he prefer the decrees of an ancient General Council,
if such there be, to the rashness and ignorance of a few.
But what if some error spring up concerning which nothing of this kind
is to be found? Then he must take pains to find out and compare the
opinions of the ancients, provided, of course, that such remained in
the communion and faith of the One Catholic Church, although they
lived in different times and places, conspicuous and approved
teachers; and whatever he shall find to have been held, written and
taught, not by one or two only, but by all equally and with one
consent, openly, frequently and persistently, that he must understand
is to be believed by himself also without the slightest hesitation.


The Sedevacantist vs. Sedeplenist Debate in the Church Today
Sedevacantists (from the Latin sede vacante = &quot;vacant chair&quot;) are Catholics who believe currently there is no pope. They generally consider John XXIII, Paul VI, John Paul I, John Paul II, Benedict XVI, and Francis all anti-popes. They argue that since these men have proclaimed error and heresy in an official manner, they, as St. Francis de Sales said, ipso facto fall out of the Church. (St. Robert Bellarmine, doctor of the Church and &quot;Master of Controversies,&quot; says the same.) What is no longer part of the body of the Church cannot be its head; thus, they are not real popes.
Sedeplenists argue that they are true popes. Although they sometimes agree these pope claimants teach error and heresy, sedeplenists do not consider them formal heretics because they judge them as lacking pertinacity in proclaiming error and heresy or as being invincibly ignorant. They say only a future pope or council can depose someone who could be an anti-pope.
There is also the material pope thesis, which essentially says these pope claimants are more like kings than popes.

The Church's Theological Notes or Qualifications
Listed below are the so-called theological notes and their associated censures from the table in Sixtus Cartechini, S.J.'s 1951 work De Valore Notarum Theologicarum (On the Value of the Theological Notes), which confessors have used when dealing with erudite penitents. (It's also available in Italian translation.) The theological notes are a way of classifying the proximity of a theological proposition to revelation. (For a good history of the development of these notes, see The development of the theological censures after the Council of Trent: (1563-1709) by John Cahill, O.P.)
There is only room for &quot;reasonable interpretative variation&quot; in the lowest of the notes (#9 and #10). As Pope Pius XII wrote in Humani Generis:

Popes generally leave theologians free in those matters which are
disputed in various ways by men of very high authority in this field;
but history teaches that many matters that formerly were open to
discussion [e.g., the Immaculate Conception], no longer now admit of
discussion.

For the Immaculate Conception, the discussion ceased when Bl. Pope Pius IX defined it dogma in Ineffabilis Deus (1852).

Theological note: Dogma.
Equivalent terms: Dogma of faith; de fide, de fide Catholica; de fide divina et Catholica.
Explanation: A truth proposed by the Church as revealed by God.
Examples: The Immaculate Conception; all the contents of the Athanasian Creed.
Censure attached to contradictory proposition: Heresy
Effects of denial: Mortal sin committed directly against the virtue of faith, and, if the heresy is outwardly professed, excommunication is automatically incurred and membership of the Church forfeited.
Remarks: A dogma can be proposed either by a solemn definition of pope or council, or by the Ordinary Magisterium, as in the case of the Athanasian Creed, to which the church has manifested her solemn commitment by its long-standing liturgical and practical use and commendation.
Theological Note: Doctrine of ecclesiastical faith
Equivalent term: De fide ecclesiastica definita
Explanation: A truth not directly revealed by God but closely connected with Divine revelation and infallibly proposed by the Magisterium.
Example: The lawfulness of communion under one kind.
Censure attached to contradictory proposition: Heresy against ecclesiastical faith.
Effects of denial: Mortal sin directly against faith, and, if publicly professed, automatic excommunication and forfeiture of membership of Church.
Remarks: It is a dogma that the Church's infallibility extends to truths in this sphere, so one who denies them denies implicitly a dogma or Divine faith.
Theological Note: Truth of Divine faith.
Equivalent term: De fide divina.
Explanation: A truth revealed by God but not certainly proposed as such by the Church.
Example: Christ claimed from the beginning of His public life to be the Messias.
Censure attached to contradictory proposition: Error (in faith).
Effects of denial: Mortal sin directly against faith, but no loss of Church membership. May incur a canonical penalty.
Theological Note: Proximate to faith.
Explanation: A doctrine all but unanimously held as revealed by God.
Example: Christ possessed the Beatific Vision throughout his life on earth.
Censure attached to contradictory proposition: Proximate to error.
Effects of denial: Mortal sin indirectly against faith.
Theological Note: Theologically certain.
Equivalent term: Dogmatic fact; theological conclusion.
Explanation: A truth logically following from one proposition which is Divinely revealed and another which is historically certain.
Example: Legitimacy of Pope Pius XI.
Censure attached to contradictory proposition: Error (in theology).
Effects of denial: Mortal sin against faith.
Theological Note: Catholic doctrine.
Equivalent term: Catholic teaching.
Explanation: A truth authentically taught by the Ordinary Magisterium but not as revealed or intimately connected with revelation.
Example: Invalidity of Anglican Orders; validity of Baptism conferred by heretic or Jews.
Censure attached to contradictory proposition: Temerarious.
Effects of denial: Mortal sin indirectly against faith.
Remarks: The expression Catholic doctrine is sometimes applied to truths of a higher order also, but never of a lower one. In some cases the appropriate censure may be graver than &quot;temerarious&quot;.
Theological Note: Certain.
Equivalent term: Common; theologically certain.
Explanation: A truth unanimously held by all schools of theologians which is derived from revealed truth, but by more than one step of reasoning.
Example: The true and strict causality of the sacraments.
Censure attached to contradictory proposition: Temerarious.
Effects of denial: Usually, mortal sin of temerity.
Remarks: Proportionately grave reason can sometimes justify an individual who has carefully studied the evidence in dissenting from such a proposition; since it is not completely impossible for all the theological schools to err on such a matter, although it would be highly unusual and contrary to an extremely weighty presumption.
Theological Note: Safe.
Explanation: Affirmed in doctrinal decrees of Roman Congregations.
Example: That Christ will not reign visibly on earth for a thousand years after Antichrist.
Censure attached to contradictory proposition: Unsafe/temerarious.
Effects of denial: Mortal sin of disobedience and perhaps imprudence.
Remarks: Exterior assent is absolutely required and interior assent is normally required, since, though not infallible, the Congregations possess true doctrinal authority and the protective guidance of the Holy Ghost.
Theological Note: Very common/commoner.
Explanation: The most solidly founded or best attested theological opinion on a disputed subject.
Example: Antichrist will be of the tribe of Dan.
Censure attached to contradictory proposition: None.
Effects of denial: None.
Remarks: Very common or commoner opinions can be mistaken and there is no obligation to follow them though prudence inclines us to favour them as a general policy. It should be noted that an opinion which is &quot;very common&quot; is less well established than one which is &quot;common&quot; which implies moral unanimity of theological schools.
Theological Note: Probable.
Explanation: A theological opinion which is well founded either on the grounds of its intrinsic coherence or the extrinsic weight of authority favouring it.
Example: Judas received Holy Communion at the Last Supper. Judas did not receive Holy Communion at the Last Supper.
Censure attached to contradictory proposition: None.
Effects of denial: None.
Remarks: The better founded of two conflicting opinions is referred to as more probable; but Catholics are free to prefer some other opinion for any good reason.