in my opinion it's not necessary to have a car, but it's pretty damn convenient.
i lived in santa catalina last year, and it is completely possible to do everything without a car. most likely you'll either bike/walk/take the bus to campus daily. also, the bus runs right up to camino real market place, so you really don't need a car to get to a grocery store or the movie theater or some local restaurants. the bus also goes downtown, so that takes care of going there. to get to la, you can always take the train or the greyhound which leave from downtown sb. (plus, you don't know anyone now, but trust me, by the end of the first few weeks, you'll have tons of friends

) if an emergency does arise, you can almost guarantee you will know somebody with a car to help you out. and the only thing i can think of that you may need to take more of with you is clothes, since you won't be able to go back and forth to home as often.
like i said, it's obviously way easier to have a car, but if for some reason you can't/decide you don't want to bring it, you can still do everything.