Use a video capture device, not a DVD recorder
I've converted a VHS tape to video files using hampshireman's 'standalone DVD recorder' technique, but wouldn't recommend it. It's a whole bunch of extra steps on top of the methods recommended in the previous posts, and it's using already antiquated technology. DVD recorders are indeed getting hard to find, and support for them will already be limited.
Also, my DVD recorder ripped to 576i format, whereas purpose-made capture devices might manage the superior 576p.
For transposing the rest of my VHS collection I'll be using the methods described above in this topic (thanks posters!).
As the others have stated, this requires buying a capture device, which converts a composite VHS signal to digital input for your computer, and has two (i.e. stereo) audio input plugs.
bassfisher6522 posted a link for a very nice looking capture device which also has a DV input for more modern devices. However video capture devices with this added feature appear to be rather rare. All you're likely to need is composite video (the one yellow RCA connection) - all video capture devices should come with this. In the very unlikely event your VCR lacks an RCA composite video out socket and two (stereo) RCA audio sockets, you'd have to get a different device and/or cables.
About any device like these would probably do the job:
eBay U.S.: video capture devices
You'll also need RCA plug to RCA plug cable(s) of suitable length to connect your VCR player to the capture device.
Personally I'll be paying above the minimum for a good brand, and if there's any software bundled I'll be checking it works on Windows 7 before buying. But you can use the freeware mentioned in this topic. I've installed ScenAlyzer, which pmat recommended, and it looks very nice. Bundled programs are sometimes awful anyway.
If your computer already has a TV tuner card, it's possible it has composite video and stereo audio inputs which could negate the need to buy a video capture device.
Happy transposing. Like you I'm looking forward to freeing up a lot of shelf space currently occupied by VHS tapes!