Delux,
It almost sounds like you are working off of GEN power as opposed to the battery. A fuel problem usually isn't instantaneous but an electrical one can be. Does your car have a ammeter on the dash...does it show the system is charging?
Your six volt starter isn't as prone to the hot start problems you hear about on the V-8s. It wouldn't be out of the question for the battery to regain a charge hefty enough to start the engine, especially if the rig had time to cool off a bit. What shape is your fan belt in? If you can start the car, watch the ammeter, turn on the lights and any other items that will draw some current...what does the ammeter do! It shouldn't swing to discharge and STAY there.
It's possible that your battery is near death and will not hold a charge. Without the proper load tester or hydrometer, you would need some parts store to check it for you. Good luck.
Stuart