If the Internet should happen to go down, there will be no access to Bitcoin until the Internet comes back up. It won't dissapear, it just won't be there while the net is down.
And in the case of nuclear war, or WWIII, food and clean water will be king like cash used to be. And people will really kill for that kind of "money."
As the old saying goes...
Cash is King... Also,
Don't put all your eggs in one basket.
It's hard to anticipate such things as a global war taking out all communications and thus access to the block chain, if someone were in a remote location, they could
in theory I guess create multiple transactions for the same balance of BitCoins they have in their possession. But, then again, how would you know unless it were something like a paper wallet?
Above all else, make sure you have a nice deck chair, a bottle of something red, a bottle opener, some blue cheese (and some matches) and sit on a nice hill top and watch the red glow as the nuclear missiles stream overhead.
... I'll drink to that!