Do you have a video ipod, iphone or any phone or mp3 player that plays video? If so, there are ways to "rip" the DVD and make a copy to transfer on to that device. That would probably be your best bet, and the other advantage is that, if someone wants to watch something else in the gym, you can still do your routine without bothering anyone else.
Now before anyone starts questioning the legality of ripping a DVD, it is legal, IF THE DVD IS ONE YOU OWN. Don't rent a DVD and rip it. Don't borrow a DVD and rip it. Don't steal a DVD and rip it (don't steal DVDs in general). Making a copy of a DVD you own is considered legal under "Fair use" copyright laws, much the same way it is legal to make an audio cassette of an album or CD you own (what's an audio cassette? Shut up, that's what)
There are programs you can buy to do this job, but there is also "Freeware" I've used and I've never had a problem with them. I won't post them here, but feel free to send a private message.