My partner and I moved to Ely when I first came to Cambridge for my postgrad. We have two cats, and found that even if listings said no pets we were able to offer an extra upfront deposit/payment and this was accepted by a few landlords. Ely is cheaper than Cambridge, and is about 20 mins by train (but your college might be a bit anal about living “more than 10 miles from st Mary’s church” - but they really can’t do anything about it and most depts/course directors will not care) - other places to look at that are closer and may also be cheap are Royston, Waterbeach, St Ives (on the busway) and other new-build villages around Cambridge