My dogs were essential to me staying sane in residency. I had 3 (4 now), but I also had a layabout, chronically-unemployed alcoholic husband whose only real function was as a dogsitter. (He didn't start out like that, but that's how he ended up.) The dogs were a big reason why I stayed with him, as it wasn't much of a marriage. Also, residency would have been a crappy time to try to separate, and honestly, my dogs were more important. Heck, I figured that if he wasn't doing anything else, he could help with MORE dogs, and we volunteered with a local rescue as a foster family. I do love dogs!
I am now done with residency, in the middle of the divorce, and my dogs have been a huge rock for me. They make me exercise. They love me unconditionally. They don't smell, or drink beer excessively, or waste money. (Not that I'm ranting about a certain ex-life-partner, or anything) 😉
One of the OB residents who started when I did had a dog, but lived fairly close to campus. The other OB residents sort of adopted the dog, and took turns walking him when she was on call. That might be an option. A roommate might be as well.
I think with dogwalkers and the right dog, it would be doable. It might be a lot of work, but since when do good things always come easy? If I were in your shoes, I'd probably contact several rescues and start thinking about a slightly-older (3ish) mixed breed. Or a retired racing Greyhound. Also, as a woman living alone, there's something to be said for having an alarm system with teeth. (My German Shepard is more bark than bite, but he's a hell of a lot of bark)
Just my dog-lovin 2 cents.