Chris,
Had a similar problem with my '54 3/4 ton.
Drove me nuts! :mad:
Right after it happened for the fifth time, I was sure that the Gas gauge was lying and that the tabk was empty, so I took off the gas cap to put in more from my 5-gallon can, and when I loosened the cap, I heard this rush of air... going INTO the tank !

The vent in the gas cap wasn't letting air into the tank, so after a few moments operation, enough "vacuum" developed on the tank to prevent any more fuel from getting to the carb.
Try the follwing: remove or at least loosen the gas cap and try running the car, and see what happens.
If the problem persists, then I would begin to suspect either collapsing rubber lines, enough dirt in the system to prevent the fuel-pump valves from functioning properly, or...
the fuel "pick-up" pipe in the tank is restricted with rust (I've seen 'em block-up solid!). Don't know if the car has a
"top-feed" tank or not, but if any part of the plumbing from the tank to the pump is restricted, that can give you grief!
I have taken to putting a disposable Fram fuel filter on the inlet side of the pump, to catch any tank crud before it can screw up the pump or carb.
I also carry a few spare filters in the vehicle in case they plug-up on the road.
Good luck!