Unless your 60 has a Holley Sniper or some other fuel injection on it it's going get fuel via a mechanical fuel pump on the passenger side of the block.
I’d start by checking the following:
-Are there any blockages in the fuel line that goes to the carb?
-Any blockages in the fuel lines going to the pump?
-Is any fuel leaking out of the fuel pump or it's connections?
-Check the oil and see if it smells like fuel because the internal fuel pump diaphragm is leaking internally and pouring into the crankcase
Here's a good article on them:
If a car stops and has not run out of petrol, check whether there is petrol in the carburettor . If there is not, there is a fault in the fuel system .
www.howacarworks.com