There are various possibilities where you could live in Göttingen.

1. Dormitories. There is a long waiting list. If you want to live in a dormitory, you need to apply at least 6 month before your arrival date. Online application is possible.
Further information is provided here: Dormitories Studentenwerk Göttingen

2. Private housing market. There are various websites offering announcements.
studenten-wohnung.de
studenten-wg.de
wg-gesucht.de
zwischenmiete.de

3. Some other helpful advice is provided here:
Finding a Place to Live
Accommodation for students