Trouble diagnosing problem

What kind of gas cap does it have? It sounds like either it has a non vented cap and should or it has a vented cap that's malfunctioned and doesn't vent anymore. Take the little gas can with you full. Drive it until it quits again. When it does, Remove the gas cap and see if it will restart. If it's pulling a vacuum on the tank from not being vented, it will stop the fuel flow and you will indeed "run out" of gas. Try that and report back. If it works, get a new vented gas cap.....and also look for the factory fuel tank vent. It could be stopped up.