It is possible to live in Los Angeles without a car but it is best advised to have a car if you are able to. I lived in Los Angles for four years (the entire time I lived there) without a car. As a female, there were some scary moments. I missed the car mostly for security/safety reasons.
Los Angeles has changed since I have lived there in 1983 to 1987. The bus system was great than - frequent and 24/7. I don't know what it is like now. The passengers , however, posed certain challenges and I was once punched in the back by an ill woman who said I stole her clothes. Riding the bus in L.A. was an overall bad experience, but I learned where the safer bus stops were and didn't walk at night too much. You will be walking a lot to catch a bus because , of course, it's not door to door service. I recommend you get used to lots of walking before hand. The L.A. bus routes are super long and cover a large distance.
I was less apprehensive without a car when I lived in NYC. I was never bothered on the streets in NY like I was in L.A. at that time.
One thing I had in my favor was that I had a lot of friends when I lived in L.A. and they had cars. I don't feel like I missed out on anything being car-less. There was always a way to get where I wanted to go.
Another thing to consider is that California may have one of the highest car insurance rates. You will not only have to pay for gas and maintenance for a car, but will also have to pay for car insurance which is high priced in general if you want to be strongly covered.
In conclusion, my answer to your question is that you will be able to live without a car in Los Angeles but it is highly recommended to develop a plan to buy one.