I've done this;
It ran a webserver for testing etc etc, an internal fileserver, and an external facing bitorrent. it also gave me a testing platform for Linux.
I used Ubuntu server (http://www.ubuntu.com/download/server/download
The downsides are you have a box running 24/7 and cheap computers / computer parts don't like to do that. I had to have a monitor, keyboard and mouse for set up but once I had set it up I used SSH to get on there and thence install Webmin
wich allows control via a web interface most things.
I like the idea of a VNC setup though, now, I just use a NAS box that promised to do webserver but doesn't really it does do bittorrent though, so....
There is also, if its just a "NAS from old PC hardware" scenario, you want...http://www.freenas.org/