Actually, though, we don't even need to consider the apocalypse. The fragile state of digital storage is already causing trouble. NASA has a few people racing to recover old images from its Lunar Orbiter missions in the 1960s, which are currently stored on magnetic tapes and may not be long for this world. And the National Archives is struggling to preserve its digital records, which tend to rot faster than paper records.
A related tale of disintegrating media comes from Larry Lessig's Free Culture—though this one has a twist. There are a lot of films that were made after 1923 that have no commercial value anymore. They never made it to video or DVD; the reels are just collecting dust in vaults somewhere. In theory, it shouldn’t be too hard to digitize these films and put them in an archive. But alas, thanks to the Sonny Bono Copyright Term Extension Act that was passed by Congress in 1998, any film made after 1923 won't enter the public domain until at least 2019.
That means these films are still under copyright, and anyone who wanted to restore them would have to track down the copyright-holders (not always easy to do) and probably hire a lawyer. And who's going to go through that much trouble just to restore some obscure movie that only a few people might ever watch? Yet a lot of these older movies were produced on nitrate-based stock, and they'll have dissolved by the time 2019 rolls around, leaving nothing behind but canisters of dust. It's sort of tragic.
It's also a perversion of the original intent of copyright laws. Copyrights like patents are government imposed monopolies that dampen commerce and development of new works. Intellectual property rights were seen, in the words of Jefferson, as a necessary evil to balance the interests of the creators with those of the general public by granting temporary these monopolies.
The suggestion that extending these monopolies for almost a century is meant to protect the interests of creators is absurd. The vast majority of these rights are held by companies like Disney or Time-Warner, companies that frequently screwed over the original creators and are now spending more money lobbying to keep the rights than they did to actually acquire them. This is particularly egregious for Disney, a company founded on adaptations of public domain works.