Any old computer can be used for simply serving up media, and running other useful stuff like MySQL (for centralised XBMC database), and various other server-type applications.
My best recommendation isn't hardware - but software. Having a way to easily extend your storage, or upgrade older disks, will be the biggest difference between having something that makes your life easy, or complicated.
I personally use Windows Home Server 2011 (for the in-built automated backup of all the computers in my house), and then on top of that, I use
StableBit DrivePool
This allows you to add one or more hard drives, of any type, any size, and any connectivity method, to a single "storage pool". This means that I have a single "folder" which contains all of my server's storage - and I don't have to worry about which folders go on which drive, and moving stuff around if a folder gets too large. I can also swap out a small disk for a larger one with only a few clicks of the mouse.
It doesn't matter that my largest physical drive is 3TB, but I have a folder which is 4TB in size - when I'm browsing that folder, I see
everything within that folder - they may be physically stored on 3 separate drives, but I don't need to go hunting for it.