I'm still not solid on what I think about private companies having the responsibility to provide your data for you if they're closing down, but his bit at the end about the archive team seems... well, brilliant. Hell, it could even be a paid service. Monitor the web for companies shutting down and download away. Of course, that's assuming that the data's not protected behind some sort of wall (if Facebook was closing, or something like that); in that case, contact the company and try and make a deal. You could get them to simply give you all the data, or (in case of privacy concerns, etc.) offer the closing company a solution where they commit to letting users get at their data for 90 days (or some similar number) and after that, you put up a page where people can pay to access the original site, or at least some sort of download page.
Of course, people could just back up, but some stuff is just hard to do that with, especially on the web.
Of course, people could just back up, but some stuff is just hard to do that with, especially on the web.