What JanT says is true.

I had a martial arts teacher who used to talk about taking care of number one. He said, if you don't take care of number one, you're actually letting down all the people who depend on you, because eventually, you're going to crash, and when you do, they won't have you to lean on.

So take care of yourself. Do what it takes. I know lots of people who ride on-call with a pager; surely no one expects you to just sit by the phone waiting for an emergency? Can you just bike around your local neighborhood, so that you're close to your car in case something comes up?