What is the minimum infrastructure necessary for Bitcoin to work? Let's say there's some sort of end of the world situation. Everyone's computers still work, but the Internet does not. Would Bitcoin still "work"? Could two people transfer Bitcoins to each other directly?
In case for some reason the Internet stops working, Bitcoin could be incorporated into any transmission medium with enough throughput to at the very least support block propagation. This would require ideally 1MB of data to be propagated through the globe in under 10 minutes. The bare minimum might make it very hard for Bitcoin to work effectively, but media such as radio, telephone or the like could probably handle the Bitcoin load without much problem. Synchronising new people to the Network might be very resource consuming, but still doable.

In case the traditional means of sending information through the globe would become impossible, the Bitcoin Network would become too fragmented to be reliable. While one would still have access to one's Bitcoins locally, one could never be sure if they are dealing with the longest blockchain, thus double-spends could be a serious threat for any transaction, no matter the (local) confirmations.

In such scenario only dealing with Bitcoins that are already stored offline if form of physical temper-proof currency could be useful - Casascius coins for example. One could expect, however, that the coins would depreciate in value along with the exchange rate of (now useless) Bitcoins.

All in all, as long as you can send information around the globe at a decent speed, you can still use Bitcoins.