This is a proposal I will probably implement but welcome feedback on.
I do not really like git for synching/backup, because

1. Most free git hosting services (especially with free private repos) are designed for developer. Usability for "ordinary" users is really questionable.
2. People need yet another account. In contrast, several email and hosting providers offer webdav just as part of accounts people might already have.
3. Git keeps a history (and makes removing parts of it really hard). For security reasons I don't think leaving old passwords around - or worse still current password encrypted with a now-compromised key - is a good idea.

The only alternative that seemed easy enough to support is WebDAV.
Windows offers the WNetAddConnection3 function, which can basically do everything, ask for username, password, pick a drive letter and map it there. It can then be used as normal files.
Linux has davfs2. This is more tricky, as we will have to read in the password and pass it to the mount command, plus if it is not configured suid then only root can mount etc. But it should still be possible with reasonable effort (and would be easy to extend to use sshfs or anything else instead).
I have no idea about whether this is possible to do on OSX, too.
Unless someone has better ideas or convinces me this is a really bad idea I'll probably try to implement these and see how well it works.