I'm looking at building one of these myself. If you're comfortable building your own machine, then basically any computer with enough horsepower and memory to run video editing software at a reasonable speed will do, as long as it's equipped with the right add-ons.
You'll need:
A video capture card if you want to pull video from analog sources (VCR, Cable TV, that sort of thing). ATI's All-In-Wonder line of video cards has this built in, and they also sell an add-on "TV Wonder" card that does this. Hauppage also has a line of these. A card like this will also basically act like a TiVo. An add-on card will run you between $50 and $100, depending on feature set. The AIW cards are priced $50 or so more than the equivalent 'plain' ATI video card (so, from about $150 for the AIW 9000 to about $400 for the AIW 9800 Pro)
A Firewire (IEEE 1394, also called 'I.Link' by Sony) card (or capability built into the motherboard) if you want to grab *digital* video from a digital camcorder. These are cheap, and built into most new motherboards anyway.
A DVD reader to pull video from DVDs (or a writer, see below). Breaking copyright encryption on DVDs is illegal! But it's not like the MPAA is going to hunt you down unless you start reproducing them and selling them on the street. I'm not entirely sure that this portion of the DMCA would even hold up under fair use copyright law (for personal use), but that's a different issue. Regular DVD readers are about $50.
You'll need a DVD burner to burn or copy DVDs. They run between $150 and $300 or so, depending on features and speed.
The base system would be something like this:
Intel or AMD motherboard (probably i865/i875 for Intel, or nForce2 for AMD -- between $100 and $200).
Intel or AMD processor of at least 2Ghz (Intel -- Pentium 4 2.4C for ~$150, AMD -- Barton 2500+ for ~$100)
At *least* 512MB of DDR400 RAM. More will make working with large files easier. $100-250, depending on amount and speed.
Hard drives to taste (I'd recommend at least 200GB total for a video recording/editing box -- disks run about $1/GB now, sometimes less with rebates). It might be worth setting up RAID if you work with very large files on a regular basis, and/or getting some fast 10kRPM SATA drives.
Video card, with video-out (I'd go with ATI or Matrox for better 2D output quality). An ATI All-In-Wonder card will also do video capture, and comes with editing software. Runs from $50 to $400, depending on speed and suitability for gaming. If you don't plan on gaming on this box, get a low-end card.
Most OEMs (Dell, Compaq, etc.) will sell you a system like this for $2000 or more. You could build it yourself easily for $1000-$1500. There are many sources for parts -- Newegg comes highly recommended among online retailers, although there are many others. Pricewatch is a good place to start once you know which parts you're looking for, to make sure you're getting decent prices. Just be sure to check out the store on a site like resellerratings.com before you give them your money!