Yes there have been some proposed ideas such as FileCoin, Storj and probably others. However, I'm not sure how well these systems work because there would need to be a way to prove to other peers that you actually are serving the piece of data. Some call it proof-of-storage, others proof-of-retrievability. But it's still an open problem that nobody seems to have solved yet. I mean, this is probably why both projects I mentioned above aren't ready yet.
From a technical point of view the idea of a P2P storage system is that if you upload a piece of content on this cloud, you'd expect it to be there all the time. That's the whole point of a cloud; you upload it, and can download it from anywhere in the world at any time. I'm not sure that P2P is a good solution to this issue. A client-server model, from an entity that you trust and pay directly which has reputation (say Dropbox), is probably a better mechanism of doing this.
In a P2P system, all the nodes that were holding your data could simply go offline for no particular reason and your data would be unreachable. Furthermore, even if the nodes were incentivized to keep sharing your data (perhaps being compensated through some form of cryptocurrency), there's still no way to figure out how the incentive would work exactly. For example, how would you stop yourself from serving your own data using another computer, and therefore getting all the crypto yourself?
It's a very interesting problem and I think we're slowly getting there, but we really need to get the incentives right, and it's a really hard problem in a P2P system. But hey, there's hope, just look at Bitcoin!
2The blockchain has to be stored by every full node, so it's not exactly the best place to store your data (good redundancy but no one wants to store everyone else's data). The Storj project that Luca pointed out uses a blockchain to facilitate payment when serving up files, but does not store the data in a blockchain itself. You may also be interested in Factom, they use the Bitcoin blockchain to secure data from modification. – morsecoder – 2015-04-18T11:48:18.410
so who stores the data? if users did, what if one of them turns off their computer? or a bunch of them do at once? – Patoshi パトシ – 2015-04-18T14:48:21.403
You have to pay multiple people to store your data in case one of them isn't available to serve it up. – morsecoder – 2015-04-18T15:53:16.627
thats another thing how long do they store it for per each payment? – Patoshi パトシ – 2015-04-18T16:45:29.273