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For backup, you could use duplicity or obnam, both of which gpg encrypt incremental backups.

For file storage, you can use git-annex which gpg encrypts data stored on rsync special remotes. I hope to add gpg encrypted git repositories to it soon too.

Or tahoe-lafs, which encrypts files and allows sharing them by sharing a special url.

What? Encrypt it on your system, store the encrypted blob there. Add a simple web server to host them.

If you do any crypto on these servers, obviously anyone with physical access could compromise it (although potentially not without alerting you that it had been).

Set up VPN...