You will have to experiment to find out the answer to that one.
They used to have mowers that you wound up and pushed a button to start them, the springs were about an inch wide wound up in a four inch or so shell. Finding something like that would be a good place to start, They would spin a 2hp motor for about 6 seconds if you gear it right you might get a minute or so out of it. Keep in mind it all about how much torque you need to drive the gear train and transmission and how much torque over what time the spring will provide. You will have to build a gearbox/throttle system to transfer the power from the spring to the wheels, and provide enough power to drive the car. Look at it this way: All you want to is build a watch with a transmission that will spin wheels, not hands forward, back, stop and go (using electronics for control).
Interesting project, a lot of work, but it sounds like a neat thing to try.