I got one of those Psion Wavefinder DAB radio tuners for my PC (under fifty quid) and with the (free!) software you can download for it, you can tell it to "tape" things on a particular channel at a particular time etc. (provided the computer is switched on!)

I would imagine someone has written something for capturing the audio stream from BBC Radio 7 broadcast over the internet, and if you find it, please let me know as that would be very useful!