There are a couple of possibilities, both hard to find to be sure. There is an over limit Klixon thermostat like device inside the heating section of the furnace. When the air temp goes over the pre set limit of it, it turns the burner gas off. They are somewhat unreliable, especially on older units. They cool down and should re establish contact, but don't, and the furnace will not fire properly on the next start up. The second possibility that comes to mind is the relay inside which controls various functions. They can get old and tired too.
I don't think you have any thermocouple issue, as your pilot light remains lit. Your sail switch is working, or you would not have heat at any time. What I don't understand, because I have never seen or heard of the problem is that "The fan might turn on for 20 or 30 seconds, but then it tuns right off". I can think of no reason for that on the old non electronic ignition furnace you have. (oops, on re reading prior to posting, I thought of a reason, could be the anticipator in the thermostat is set too "fast" and all you are getting is a relatively quick heat demand then the thermostat turns off from the anticipator)
I know I can sound like a broken record on this issue when I post, but any 38 year old furnace can have so many issues, some safety related, that you might consider replacing it with a new one. Often you fix one thing and another crops up. If you are not big on taking the furnace out, working on it, putting it back in, having it work (maybe) and repeat, think about replacement. On those years and models of Suburban Furnaces there was a long standing recall for a defective crossover tube which was fixed in a recall. However, if yours was not fixed, it is still a safety issue, and Suburban no longer has the recall parts available, nor do they support any parts for those furnaces.
A new furnace will have electronic ignition, so no more pilot lighting. It also will probably have a somewhat lower current draw from the fan motor, good if you camp without power available.