You have quite a few options to consider, for both your price range and usage.
As you say this is going to live next to your TV you like something small and in-audible.
I imagine (as you already dont have a server) that your website is having heavy usage, so nothing ludicrous is needed.
If you want to go prebuilt, there are a couple of options. Lots of people I know have had great success with the HP microserver. The older NL54 (G7) are pretty cheap and feature a dual core Turion and 4GB of RAM. With space for 4x 3.5 HDDs. Pretty much the most compact thing you can get and they are pretty damn cheap atm. In the UK one can be bought new for £130-ish.
If you want something a bit newer and a little more flashy in the aesthetics then there is the newer G8 HP microserver. At this time I do not know if a small firmware bug, that makes the fan ramp up under linux systems has been fixed yet. But I do not know how loud it is as I have not gotten hands on with one.
The more appealing option to you, I imagine, is to build it yourself. On the Intel side for both small and low power consumption is the Atom C2750. An octa core Atom based SoC, ASRock make a very nice ITX form factor board featuring this chip. With plenty of SATA connections for all those HDDs and the nice feature of IPMI (don't know how useful this would be to a non-server guy) which allows for remote health monitoring and statistic viewing. There is a slightly cheaper quad core version that uses the Atom C2550, although if you want a lot of room for future expansion or headroom for VMs (or something) then I'd recommend the C2750.
AMD has some nice options as well, you could use an AM1 based system with either a Dual or Quad Core Sempron/Athlon. The main benefit for this would be cost, although you will lack things like Dual Gigabit LAN and the IPMI. Also the chipset only supports 2 SATA connectors (which would be an issue for later expansion or just general use out of the gate). This could be solved with an add-in/RAID card.
Then it would simply be a case of finding a case (get it lol) that you like and fits your requirements.
The reason I suggest the HP Microservers, the Intel Atom and AMD AM1 is because they are all low power and as a result don't require massive amounts of cooling. Hell the Atom system is designed to be passive!
My previous suggestions are all about getting new hardware. Finding old/second hand server grade stuff (although cheap) will be inherently larger, use more power, produce more heat and noise. Although looking at i3's in a small ITX board is also an option.
I suggest you go off and come back with some hardware idea's and we will give our opinions and further suggestions.
- zanginator