For your purposes, one of the lower end dells would work just fine. They're not silent, but they are *very* quiet.
If you want to build it, you need to understand that for what you want, the via board and processor mentioned by FishTankX would be an excellent choice. Their 1gig processor definitely has the power to do what you want. Add a fanless zalman heatsink, or an aavid cooler from siliconacoustics, for example. buy some fan isolators and one of their ultra quiet 80mm case fans, while you're there. the 80mm panaflo will operate at 7v rather than 12, if it's too loud for your tastes.
I only know of one vendor for the right case/psu combo- casedepot carries the liteon st050 w/ ultra quiet seasonic psu. It's plain jane, but it breathes well, and may not even require a case fan with the via system. You don't need 300w, but you do need the ultra quiet fan, and this is one of the better buys. Beware of cheap psu's- they make up for lack of beefy heatsinks with lots of airflow using noisy fans....
Check storagereview.com for quiet hdd's- a 5400 rpm model will do what you need. Several outfits sell the silentdrive enclosures, or similar, if you want to spend the money for one. Most 7200rpm drives run too hot for these enclosures.
Getting significantly quieter than this will cost mucho dinero- Check out the fanless psu from siliconacoustics, for example, and solid state storage drives are available, at exorbitant cost..... That Hush ITX looks nice, except for the price.....