We have a Minerva real-time analog to DVD-compliant mpeg converter sitting on a shelf.
It has video & audio inputs and connects to a computer. There's an interface for the device which allows you to configure settings for the bit rate and such.
Basically, it's built for studio production and video to satellite broadcast for the big networks, cable TV, etc. You can find them on eBay for about 500-1000 bucks now depending on the unit's features (10 years ago they were about $15K, but HD video became popular and the Standard Definition VHS and DVD went out of fashion soon after). Anyhow, for SD video, it's a very high quality professional / industrial device. I recommend them if you have a lot to transfer and want top notch image quality.
ETA I think we still have some pinnacle and AJA cards that fit into PCI slots, which I seem to recall doing pretty good conversion work ok too.
If you're going to do it yourself on a shoestring, I'd recommend taking a look at one of these:
Pinnacle Systems Video Capture Cards. You can find them for less than a hundred bucks and they will perform the job. They're not a high-end device, but if you just want to make a cheap digital back-up, the kit is set up to connect with standard, consumer grade VHS outputs (like Composite Video (RCA or S-Video (mini-DIN) and RCA audio jacks). The more sophisticated devices offer high-fidelity input and output, plus allow deck control. Professional grade capture cards tend to be for editing of things like TV shows, commercials, etc. Industrial converter boards and appliances are usually a bit more specialized and limited in function, but ones like the Minerva I mentioned offer the same mpeg quality as the production companies who stamp out the DVD's you buy commercially. Minerva or Optibase converters are just a couple examples of the type of many of them use. Depending on what you're willing to spend, how much video you have to capture and the quality you want in the end, you might be able to put a nice dubbing system together for SD video pretty cheap because most of that gear has gone way down in price in the last few years.
Good luck. I hope you can enjoy your videos for a long time now.