CSU is right in downtown Cleveland. However, Euclid Avenue is in the midst of a major infrastructure redo...which means the area is a mess right now

Once they're done with this, I think it will be very nice.
Welcome to The Euclid Corridor Transportation Project
There is some student housing on campus but not much in the surrounding neighborhoods. CSU has primarily been a communter school. The closet reasonable area would be the growing Asiatown.
AsiaTown Cleveland
However, Cleveland Hts., Little Italy, Lakewood are all within reasonable distance(20 minutes) and have access to public transportation.
Greater Cleveland Regional Transit Authority
I prefer Lakewood. It's 10 minutes by car to CSU, has great freeway access, is close to the lake, has some decent little shops and restaurants and is reasonably affordable.
CSU is relatively affordable. I don't know your situation but there are grants, financial aid etc... as well.