My home server runs on Ubuntu Server 12.04 LTS (there is no GUI).
I would recommend a Linux installation, Mint works well but you can also install Ubuntu if you feel more comfortable (doesn't have to be server edition).
Depending on how you want to set up your backup, you will want Samba shares and SSH (with keys).
That way you can automate the whole thing.
If you are interested in going the Linux route, I recommend checking out Jayce's tutorial for his home server in the Linux section. Really interesting read, very detailed.
EDIT: Here is the link, it is mostly for the installation of Rsync and SSH to do the actual copying but it will give you a great starting point. (props to Jayce)
HOW TO - Back up your data using Rsync and SSH.