DIY remote back up?

I would like to put a computer at a friend's house, and then back up my local computer to that remote computer. I have access to their router, so I can set up port forwarding as necessary. I would want the back-up to be scheduled and automated, and encrypted. Is this possible? Something that already exists, surely?

Try Goodsync. Not sure if you're talking about a fully bootable version of your computer at a remote site, or just your important data files. If the latter, Goodsync is a great app.

Macrium Reflect puts out a free backup that seems to work well.