Seems like you want full nodes to provide non-outsourceable proofs of custody (of blockchain state). The file-storage coins (Filecoin, Storj etc) have the same problem but for user data. I also remember Sergio Lerner had a devcon 3 talk about this idea of paying full nodes to store blockchain state.
As for specific protocols, Justin’s one at Enforcing windback (validity and availability), and a proof of custody is the only one I’ve read in detail and I think it technically works for this use-case. In fact since ethereum blocks commit to the state root and Justin’s proof of custody is constant in the size of data stored (I think) you can just pay full nodes for storing the whole state. However the overhead introduced by snarks is probably too high.
You can probably find more protocols by googling for proof of {custody, retrievalibity, storage}.