My problem is solved!
It appears that I had to replace the thermocouple. I had a spare that the previous homeowner just happened to have sitting right by the heater. I didn't really know what it was for, but when I was perusing the internet, I found a page that said to consider looking at the thermocouple when a heater won't start. Like most situations, not every heater will be the same but the function of the thermocouple is pretty standard. From what I can tell it's sort of a safety feature. One end of the thermocouple is usually near the pilot light and the other end should connect to some part of the electrical signal flow that controls a magnetic gas valve. The basic idea is that while the pilot light is lit, the thermocouple will allow electricity to flow from the power supply, through the thermostat, then through the magnetic gas valve and back to power to complete the circuit. If the pilot light goes out, the circuit is broken at the thermocouple which will prevent the magnetic gas valve from letting gas flow through with no source of ignition. My heater is about 50+ years old, so a lot of newer ones probably operate differently. But I hope this info is helpful to someone.