Any case can act as a server, so I'd measure your space and then take a look at the cases section to see what fits.
Then you need to consider how much storage you want and whether you want this in say 1 large HD or split over many HD's in raid or duplication (if your using Windows Home Server).
As for media management windows home server does that reasonably well and you can add in components (called add ins) that allow you to manage the media in better ways than the default out of the box. e.g. MyMovies (
www.mymovies.dk), Firefly for ITunes.
For downloads you can install something like uTorrent.
If you don't want the expense or WHS you could install a number of different linux clients (though I have no actual experience of them).
Also you can set up remote access to the server either using something like logmein or Remote Desktop connection so that after set up you won't need a keyboard or monitor connected to it.
I run my server with the following specs (which was basically my old gaming rig so overpowered really for a server)
E6600 Intel Dual core
2GB ram
3 * 1tb Hard Drives
1 * 500gb Hard Drive
Windows Home Server with Service Pack 3
Tsumani Dream case
Basic GFX card (can't remember what it is (but fanless was a main factor). But I have been told that you can run a WHS without a GFX card.
If you've got access to the Members Market that would be a good way of setting up a server as you can buy a lot of old kit dirt cheap that people now don't want as they've upgraded.
I use this to stream to my living room HTPC, my main gaming rig and my Netbook.
If you use WHS it will also act as a back up server and take copies of any of the PC's connected to it on a defined basis so that you can recover any PC from a point in time shoulkd it fail or you want to roll back after a problem.
Hope that helps in a small way
Taff