Accommodation Service:
For housing of students we have the service of the student union, which offers affordable flats (single rooms in shared apartments or others, just check on the website):
For general information about accommodation for students, there is a site of the university of Göttingen, which offers you a lot of information as well.
If you want to look for an apartment on your own, there are a lot of websites offering rooms or flats. As an example the probably most widely used one: