I am a full-time college professor on the tenure track. I am also a doula and work part-time as a perinatal educator. It isn't easy, but I make it work. Really, since I got the tenure-track job, I made a policy of no new doula clients- I only take repeat clients and friends. It has been good practice for me (a perpetual people-pleaser) to learn how to say "NO"...in each of my childbirth prep series, I always have at least 3-4 couples who ask me to doula for them. But I turn them down and help them find someone else. Since I instituted the only-repeats policy, I've had about 3-4 births a year. I haven't actually had to ever cancel a class due to a birth (knock on wood), but I have taught several 3 hour lectures after being awake for over 30 hours. Who knows what the students learned from those lectures!
The spring that I was doing my certification (2 years ago), I ended up doing 12 births between March and June, while working full time (never missed a class). It seemed OK at the time, but I look back on it and think
I must admit that I've realized that it isn't super good for my emotional well-being to be a doula on-call while working full time. The weeks surrounding a birth are always very stressful for me because I am constantly on edge- trying to set up my classes to be taken care of in my absence, day-to-day. Then the babies always come in the night or over the weekend! My DH doesn't think I charge enough to make up for the mental stress (and I'm starting to agree with him!) But I feel lucky to be at the births I've been at, and I feel lucky to have a job that does have some degree of flexibility.
Long story short, I think it is do-able, but it depends on your job, and whether your partner/family is supportive. I know I could never do it if I had children of my own right now.