Are they specific about the rules for no owning pets? If it's just owning, you could advertise for a dog sitting business. You don't have to train the dog, and you could have a buddy for a while, and then send them home when you got tired of them. It will also give you experience with breeds for when you can buy one of your own.
I used to "dislike" cats until one of my kids dragged one home. Turned out to be a warm loving animal. Maybe you should think twice. I think you'd be pleasantly surprised. Plus, no walking, no paying a groomer-just the mess of the litter box. But that's easy. Good luck
If you just are looking for a responssibility to have a fish is a good idea. If you want something you can bond with id suggest a rabbit. Be carreful of the breed of rabbit though because some shed more, some need more responsibility like the extra fluffy ones id do a little research to see which one is best for you!
Maybe get shared custody of a dog? Find someone who has a dog they need walked and cared for and go over to their house every day to play with the dog. It's like owning one, except it doesn't stay overnight with you. You could even actually buy a dog with someone with the agreement it stays at their house but you split food/vet fees and excercise and care of the dog.