They'd do what libraries do: collect and preserve. Several already exist, like archive.org and Google cache. But Google's cache exists for Google's purposes. Archive.org is often forced to take down material, especially since the DMCA. You need a real decentralized network that is both lawyer- and disaster-resistant.

It is impossible for, say, the New York Times to change or obliterate what it printed on some day in 1997. For NYT.com it's a simple as a mouseclick and a letter. That means there is no such thing as "public record" on the internet. Public record is the basis of a literate society.