In reply to Pj84: For the first 10 years I climbed I didn't own a car, and I managed to do loads of climbing. I used to do a lot of hitching, but unfortunately this doesn't seem to be so popular now. Although once you get into the hills, you won't have to wait for long if you have a climbing pack on your back and look like an outdoor type. It's the cities and motorways you might have problems getting a lift nowadays.
With a bit of determination and using a combination of buses, trains, hitching and walking you'll be able to get to most crags. You could also join a climbing club and team up with folk with cars. Not having a car shouldn't be a barrier to not getting out climbing. Lack of determination and enthusiasm will be your only barrier.