In the perfect world, we'd store absolutely every bit of information generated each day, but that task is almost impossible, even with dedicated efforts like the Internet Archive. Try targeting your efforts more, say just movies. You still have to find a robust, eternal-as-possible storage medium, rock-solid processes and well, making sure that medium isn't highly combustible.
There's an excellent article by Marty Perlmutter, writing for the IEEE's Spectrum magazine, that covers the enormous problems the entertainment industry faces when it comes to archiving modern cinema. The rise of digital recording has forced engineers to think differently when it comes to the long-term storage of the original data:
These days, the major studios and film archives largely rely on a magnetic tape storage technology known as LTO, or linear tape-open, to preserve motion pictures ... Housed properly, the tapes can have a shelf life of 30 to 50 years.
Go tape, right? There's only one small problem:
The problem with LTO is obsolescence. Since the beginning, the technology has been on a Moore's Law–like march that has resulted in a doubling in tape storage densities every 18 to 24 months. As each new generation of LTO comes to market, an older generation of LTO becomes obsolete. LTO manufacturers guarantee at most two generations of backward compatibility.
Damn you, tape! But the story gets even more complicated:
Up until the early 1950s, filmmakers shot on nitrate film stock, which turned out to be not just unstable but highly flammable. Over the years, entire studio collections went up in flames ... According to the Film Foundation ... about half of the US films made before 1950 have been lost, including an astounding 90 percent of those made before 1929.
On top of digital footage, most movies also involve loads of CGI (or in the case of animated features, are entirely made of it) and storing isn't a trivial matter. Perlmutter includes this interesting anecdote about the Blu-ray release of Finding Nemo, a movie from 2003:
[The studio] found that certain aspects of the original could not be emulated in its new software. The movement of seagrass, for instance, had been controlled by a random number generator, but there was no way to retrieve the original seed value for that generator. So animators manually replicated the plants' movements frame by frame, a laborious process.
Perlmutter's piece is fairly meaty, but if you have an interest in the processes and technology of storing movie history, get stuck into it.