How long are extraordinary ministers allowed to possess the Eucharist? I was reading the answers to this question on Skeptics.SE and relating it back to my own life here on Earth. I see some extraordinary ministers taking the Eucharist to the homebound and they're usually in a rush to get there.  

So, what is the maximum amount of time they're allowed to posses the Eucharist and what do they do if they find they cannot deliver the Eucharist?
The answer to the Skeptics.SE question you cite says a Presbyterian minister "supplied the communion kit (bread, wine and cup)." Since Presbyterian ministers are not valid, ordained priests, they are unable to effect transubstantiation of bread and wine into the Body, Blood, Soul, and Divinity of the Lord Jesus Christ; thus, what Aldrin had was not the Eucharist but only "bread, wine and cup."

Traditionally, only Catholic priests, whose hands have been consecrated upon receiving the Sacrament of Orders, can touch the Sacred Species with their fingers. Laypeople touching the Eucharistic Species should be a rare extraordinary (i.e., out-of-the-ordinary) exception.

Paul VI wrote in his 1969 Instruction on the Manner of Distributing Holy Communion (Memoriale Domini):


  Where a contrary usage, that of placing holy communion on the hand, prevails, the Holy See—wishing to help them fulfill their task, often difficult as it is nowadays—lays on those conferences the task of weighing carefully whatever special circumstances may exist there, taking care to avoid any risk of lack of respect or of false opinions with regard to the Blessed Eucharist, and to avoid any other ill effects that may follow.


Notice, he says Communion in peoples' hands is a "contrary usage," contrary to the traditional practice of the Church; however, he opened the door to bishops' conferences abusing this contrary usage by allowing Communion in the hand.

Thus, there isn't a law regulating what should be a very rare occurrence in extreme situations, e.g., someone is to steal the Eucharist and a layperson is the only one around, then of course he or she should touch and consume or help others consume the Eucharist as quickly as possible.

Cf. my answer to the question "How are Roman Catholics to receive the holy Eucharist at Mass?"