Looking for a small place to live for a married couple and two cats

We are a married couple with two 14 year old cats, and we are looking for a small place to stay while my wife studies her PhD in the University of Göttingen. I currently work at home in bioinformatics, and plan to apply to a PhD around here and hopefully start next year. We would like a nice place to stay considering our student-level income, and we would stay for the duration of the PhD program (approximately 4 years). We consider ourselves friendly and helpful to our neighbors, and believe that any issues can be solved with peaceful dialogue. This is our first time in Germany, so we are exited to meet new people and get to know Göttingen. My wife knows some basic german, and I am learning to speak German while we are here.