I've got two old ReplayTV boxes still hooked up to my network that I'm able to do the same thing with and it's pretty easy to back up entire shows to your hard drive and then even use your computer as secondary storage to play them back to your
TV over your network.
I know with my DVR from DirecTV it isn't possible. The shows are encrypted and linked to that specific box. You can't even pull the hard drive and watch it in another DVR unit. Basically with those units the only option you have would be to connect the DVR to your computer and capture the video that way, then move it to an external hard drive.