Some folks likely carried a tire tube under a seat somewhere. Jack the flat tire off the ground then with some tire irons to pry the tire bead off, the bad tube can be replaced. Pump the tire back up and jack the car down and your good to go after stowing the tools. Some may have just carried a patch kit.
Rims weren't detachable for most folks so poor folks learned to do it with minimal cost. If a person had more money, they might look for detachable rims and carry a spare on the running board next to the driver since there was no door there.