Maybe I'm pessimistic, but unless there's some centralized, organizing hub that assigns specific data to specific users for storage, so that it's guaranteed that all of the data is archived, I don't see it happening.
Because what will happen without such a coordinating structure is that certain data will be popular, and stored by many people while less popular, or esoteric data will have fewer willing archivers. It would lead to a situation where holes in the data develop making it difficult to access, or lost to time altogether.
I know we're just talking about the general idea, but I think this is something that will have to be precisely coordinated, and choreographed to effectively and completely pull off.
Obviously not saying it's impossible, but I don't think it's as simple as we're making out.
I think it would have to be a system that allows people to sign up to be archivers, and they basically just commit to storing data while some mechanism allocates the data without regard to its nature. So, users wouldn't be able to choose which data they archive. And that could be coordinated and executed by some decentralized mechanism.