Clearly, for any sort of data where reliable access is important this won't be viable. It seems to me that it is only useful to do distributed storage among servers, rather than laptop or desktop computers, since servers have high availability requirements already and aren't likely to be taken off the network.
Second, you probably need to trust the servers you are storing your data on. Sure, you have some guise of anonymity, but ultimately you're probably transporting over TCP/IP and so you can look at the IP sender field to figure out who's data you're storing. Obviously this is a huge problem if some malicious person decides they want to corrupt or delete all files stored by certain entities.
Both of these problems (availability and integrity) seem largely intractable when dealing with distributed storage unless you make the limitations I suggested above (trusted computers and only using highly-available servers). Definitely an interesting thing to work on, but probably not viable for consumers (though I could see it working at an enterprise level).