For use in a gym, I'd think something like a video [link|http://www.apple.com/ipod/ipod.html|iPod] would be ideal. But I think it's very limited for playback. They claim 150 hours of video playback (320x240, 750 kbps) and 20 hour battery life (for music). I don't know how you'd get your shows on there - it's probably easy if you feel like giving Apple $1.99 an episode.
There are portable DVD players - [link|http://www.newegg.com/ProductSort/SubCategory.asp?SubCategory=491|newegg] has them starting at < $100.
A portable DVD player that also records? I haven't seen any.
HTH.
Cheers,
Scott.