What do 1st century extra-biblical sources say about hell? Do we have any information about what 1st century Christians and Jews believed about hell?  I know what the Bible says on the subject, and am curious about the extra-biblical context.
Early Christians:

Epistle of Barnabas (70 - 130 AD):


  The way of darkness is crooked, and it is full of cursing. It is the
  way of eternal death with punishment. (“Epistle of Barnabas”)


Ignatius of Antioch (110 AD):


  Corrupters of families will not inherit the kingdom of God. And if
  they who do these things according to the flesh suffer death. how much
  more if a man corrupt by evil reaching the faith of God. for the sake
  of which Jesus Christ was crucified? A man become so foul will depart
  into unquenchable fire: and so will anyone who listens to him. (Letter
  to the Ephesians 16:1-2)


Clement of Rome (150 AD):


  If we do the will of Christ, we shall obtain rest; but if not, if we
  neglect his commandments, nothing will rescue us from eternal
  punishment (“Second Clement” 5:5)
  
  But when they see how those who have sinned and who have denied Jesus
  by their words or by their deeds are punished with terrible torture in
  unquenchable fire, the righteous, who have done good, and who have
  endured tortures and have hated the luxuries of life, will give glory
  to their God saying, 'There shall be hope for him that has served God
  with all his heart!' (“Second Clement” 17:7)


The Martyrdom of Polycarp (155 AD):


  Fixing their minds on the grace of Christ, [the martyrs] despised
  worldly tortures and purchased eternal life with but a single hour. To
  them, the fire of their cruel torturers was cold. They kept before
  their eyes their escape from the eternal and unquenchable fire
  (“Martyrdom of Polycarp” 2:3)


Irenaeus (189 AD):


  Christ Jesus, our Lord, and God, and Saviour, and King, according
  to the will of the invisible Father, 'every knee should bow, of things
  in heaven,, and things in earth, and things under the earth, and that
  every tongue should confess' to Him, and that He should execute just
  judgment towards all; that He may send 'spiritual wickednesses,' and
  the angels who transgressed and became apostates, together with the
  ungodly, and unrighteous, and wicked, and profane among men, into
  everlasting fire; but may, in the exercise of His grace, confer
  immortality on the righteous, and holy, and those who have kept His
  commandments, and have persevered in His love, some from the beginning
  of their Christian course, and others from the date of their
  repentance, and may surround them with everlasting glory. (“Against
  Heresies” 1:10:10)
  
  The penalty increases for those who do not believe the Word of God
  and despise his coming. . . . [I]t is not merely temporal, but
  eternal. To whomsoever the Lord shall say, ‘Depart from me, accursed
  ones, into the everlasting fire,’ they will be damned forever
  (“Against Heresies” 4:28:2)


Tertullian (197 AD):


  These have further set before us the proofs He has given of His
  majesty in judgments by floods and fires, the rules appointed by Him
  for securing His favor, as well as the retribution in store for the
  ignoring, forsaking and keeping them, as being about at the end of all
  to adjudge His worshippers to everlasting life, and the wicked to the
  doom of fire at once without ending and without break, raising up
  again all the dead from the beginning, reforming and renewing them
  with the object of awarding either recompense. (“Apology” 18:3)
  
  Then will the entire race of men be restored to receive its just
  deserts according to what it has merited in this period of good and
  evil, and thereafter to have these paid out in an immeasurable and
  unending eternity. Then there will be neither death again nor
  resurrection again, but we shall be always the same as we are now,
  without changing. The worshipers of God shall always be with God,
  clothed in the proper substance of eternity. But the godless and those
  who have not turned wholly to God will be punished in fire equally
  unending, and they shall have from the very nature of this fire,
  divine as it were, a supply of incorruptibility (“Apology” 44:12–13)
  
  Therefore after this there is neither death nor repeated
  resurrections, but we shall be the same that we are now, and still
  unchanged--the servants of God, ever with God, clothed upon with the
  proper substance of eternity; but the profane, and all who are not
  true worshippers of God, in like manner shall be consigned to the
  punishment of everlasting fire--that fire which, from its very nature
  indeed, directly ministers to their incorruptibility. ("Apology"
  48:12)


There are many more but I think these should help you.

As for Jewish sources, I am still looking for sources. Once I do, I will post them here.

I am still looking for good material from the 1st century about hell from Jewish sources. It just so happens that the concept of Heaven and Hell is a more Christian concept than it ever was Jewish and rabbis have had very little to say, if anything at all, over the vast history of Judaism (which is still a lot but little compared to Christianity). Anyway, I found this page that you will find helpful, I think:

Heaven and Hell in Jewish Tradition.

Excerpt from the above link:


  There are, however, several biblical references to a place called
  Sheol (cf. Numbers 30, 33). It is described as a region "dark and
  deep," "the Pit," and "the Land of Forgetfulness," where human beings
  descend after death. The suggestion is that in the netherworld of
  Sheol, the deceased, although cut off from God and humankind, live on
  in some shadowy state of existence.   While this vision of Sheol is
  rather bleak (setting precedents for later Jewish and Christian ideas
  of an underground hell) there is generally no concept of judgment or
  reward and punishment attached to it. In fact, the more pessimistic
  books of the Bible, such as Ecclesiastes and Job, insist that all of
  the dead go down to Sheol, whether good or evil, rich or poor, slave
  or free man (Job 3:11-19).


And then:


  The development of the concept of life after death is related to the
  development of eschatology (speculation about the "end of days") in
  Judaism. Beginning in the period following the destruction of the
  First Temple in Jerusalem (586 BCE), several of the classical
  Israelite prophets (Amos, Hosea, and Isaiah) began forecasting a
  better future for their people.   However, with repeated military
  defeats and episodes of exile and dislocation culminating in the
  destruction of the Second Temple in 70 CE, Jewish thinkers began to
  lose hope in any immediate change, instead investing greater
  expectations in a messianic future and in life after death. This was
  coupled with the introduction into Judaism of Hellenistic notions of
  the division of the material, perishable body and the spiritual,
  eternal soul.    The catastrophe of 70 CE caused a theological crisis.
  How could it be that the God of Israel would simply allow His
  sanctuary to be destroyed and His people to be vanquished at the hands
  of the Roman Empire? While the rabbis often claimed that it was the
  Israelites' sinfulness that led God to allow it to be defeated
  (mi-p'nei hataeinu, "because of our sins"), it was more difficult to
  explain why good and decent individual Jews were made to suffer.


It talks about Jewish "tradition" but that's because the Tanakh does not address heaven and hell, as we understand it, in much detail.

The above is not in any way "the" Jewish belief. Its just an article giving you insight into some of the thought and discussions of the Talmud and the rabbis.

If and when I find direct quotes, I will put them here. But I really hope this helps you in your study/research.