Can anyone tell me if there is any music store in Los Angeles where you can rent a cd, as you would rent a dvd at Blockbuster? Thank you

A library perhaps.

Most of these kinds of shops have now shut down as the legality of the service became questionable.
A library is a good option but in libarys you will not find any current music.

I agree... libraries may not have what you want, but even so... they'll probably have other stuff you forgot about or never thought about. I can check out up to 10 cds at a time at ours... its great.