You could imagine a Bitcoin micropayment protocol (https://en.bitcoin.it/wiki/Contracts#Example_7:_Rapidly-adju...) layered on a P2P network of Tahoe-LAFS (https://tahoe-lafs.org/trac/tahoe-lafs) providers competing for business.
http://www.fsf.org/campaigns/priority-projects/priority-proj...
LibrePlanet Group: http://libreplanet.org/wiki/Group:SyncReplacement
I believe you can get 80% there with git-annex or DVCS-autosync and a git repo in Tor. I believe there is a free but limited git repo hosting in Tor, but don't take my word for it.
I thought once if it would be possible to use Freenet as a replacement for Dropbox. You encrypt your files, put them in Freenet, and they are replicated there, as long as you ping them enough so they don't get automatically deleted.
http://git-annex.branchable.com/forum/tips:_special__95__rem...
It would be great if btsync supported multiple user accounts on the same daemon.