The size of the blockchain would be a huge issue here.
Yes, but I imagine the fees as well, correspondingly.
Only the "miners" would store all, nodes would be storing the headers only, or specialized server (like the equivalent of blockchain.info) would only store the header, allowing to get the file from the miners.
The question is, would this distributed file system with its own block chain be useful in anyway?
I haven't come up with a specific need, I'm under the impression it would be useful for some type of service