While computer manufacturers are now coming out with "media servers" to sell to consumers, Popular Mechanics claims that it's not necessary to buy a brand new machine. Instead, use cheap (or salvaged) parts to build your own box, as media server hardware doesn't have to be top-notch. The biggest hurdle is choosing what operating system to use. For free and fully functional, the article suggest Ubuntu. The catch: it's not that easy to configure. The other option would be to go with the expensive, albeit easy OS (Windows Media Server).
Windows Home Server had a set-it-and-forget-it appeal that required minimum management time. But we found the Ubuntu server more rewarding for the geeky hobbyist who is willing to spend time experimenting with the software creations of the open-source community. Either way, we got a powerful machine for minimal investment.
While Ubuntu is the article's Linux distro of choice, you're certainly open to trying other flavours of Linux for the same purpose. Indeed, you can certainly take advantage of what you have to build a machine that can be used for backup or storage without heavy costs.