Any old hardware will pretty much do. I run my 24x7 home server on a -roughly 8 years old!!- Pentium 4 with only 512MB of RAM.
OS: Lubuntu (LXDE) and running ktorrent, sabnzbd, couchpotato, sickbeard, Samba and NFS servers, dynamic DNS client, remotely accessible via SSH and VNC... so extremely versatile. And running smoothly, even snappy.
I am now just about to upgrade the hardware, for the sole reason that I need to run a VM on it (using VirtualBox) and this old P4 processor does not support Virtualization. If I try to run the VM, CPU usage goes to the roof. In normal operation, it hovers around 10%.
So I'm going to build a new "server" on -again- inexpensive second-hand hardware, now just with a CPU that can handle Virtualization. You might want to keep this option in mind, so you can easily grow usage on the same hardware in the future.
EDIT: Now that I think about it, if I had to deal with multiple HDD's (not there... yet) I'd install
FreeNAS in a VM for handling the storage piece alone, then maybe Lubuntu on another VM for all the apps, just because I'm familiar with the interface... hhmmm... I'm thinking out loud, but looks like my future upgrade path is taking shape...