I don’t think this will ever happen. The web is more than a network of changing documents. It’s a network of portals into systems which change state based on who is looking at them and what they do.
In order for something like this to work, you’d need to determine what the “official” view of any given document is, but the reality is that most documents are generated on the spot from many sources of data. And they aren’t just generated on the spot, they’re Turing complete documents which change themselves over time.
It’s a bit of a quantum problem - you can’t perfectly store a document while also allowing it to change, and the change in many cases is what gives it value.
Snapshots, distributed storage, and change feeds only work for static documents. Archive.org does this, and while you could probably improve the fidelity or efficiency, you won’t be able to change the underlying nature of what it is storing.
If all of reddit were deleted, it would definitely be useful to have a publically archived snapshot of Reddit. Doing so is definitely possible, particularly if they decide to cooperate with archival efforts. On the other hand, you can’t preserve all of the value by simply making a snapshot of the static content available.
All that said, if we limit ourselves to static documents, you still need to convince everyone to take part. That takes time and money away from productive pursuits such as actually creating content, to solve something which honestly doesn’t matter to the creator. It’s a solution to a problem which solely affects people accessing information after those who created it are no longer in a position to care about said information, with deep tradeoffs in efficiency, accessibility, and cost at the time of creation. You’d never get enough people to agree to it that it would make a difference.