I'm assuming you're talking about digital cable, since it's 2012.
The way this works is that your equipment tunes into a frequency over which a multiplex is broadcast. That's a digital stream that can be demultiplexed into a number of sub-streams. Each sub-stream is a TV channel.
There are standards for these things. The sensible thing to do (especially since they're unencrypted channels) is to use DVB receiver cards in a PC. I'd use Linux, but it's up to you.
You mention archiving these at "low quality". The problem here is that the channels are broadcast at a given quality, and the broadcaster has already done the hard work of digitising and compressing the content. If you want to lower the quality, you'd have to demultiplex, decompress, scale, and compress, and that's a lot of processing. Compression is especially expensive in terms of processing.
Since storage is cheap, and getting cheaper, I'd suggest storing the streams exactly as they arrive. This is what DVRs (e.g. Sky+, digital TiVo etc. do).
You have a choice between demultiplexing, and storing the streams you want individually -- or simply writing the whole multiplex to disk. Demultiplexing is cheap; but not as cheap as simply dumping every byte the tuner supplies you onto disk. Demultiplexing, of course, allows you to pick and choose which channels to discard.
I'm afraid my 2 minutes of Googling didn't reveal what the bit rate of a cable TV multiplex is - but it shouldn't be difficult for you to find out. Once you know that, it's simple sums to work out how quickly you'll fill disks, and what kind of sustained write speed you'll need.
There is free software available to drive DVB cards. One example is OpenDVB. It would be worth doing some reading around the MythTV community. Although they'll be focussed one or two channels at a time, they'll know their way around the technology, and the software is modular.
You'll need a DVB card for each multiplex you're recording, since each one can only tune to one frequency. Depending on what you discover about bit rates, it may be worth splitting the task between several computers - but in theory one computer can have multiple DVB cards.
As a vague guide on CPU power - reading from the tuner and demultiplexing take negligible amounts of CPU work. Any PC CPU of the last 10 years could do it without breaking a sweat. Encoding/compressing is really hard work. A top-end modern CPU has trouble compressing SD video in real time. To do it at all efficiently, you'd have to offload it to dedicated chips (which is how camcorders etc. manage it).