Cabin smells like exhaust

Do you notice a difference in strength of odor when the heating flaps are closed completely?  Those smells can be sucked in by the engine's fan and forced through your heating system until they come out in the passenger compartment.

There are a number of places to check, starting with the integrity of your exhaust system.  That includes checking for leaks where the exhaust manifolds bolt up to the heads (front and back), muffler attaches to heater boxes (donuts) and the peashooters.

You should also check the condition of your heater boxes.  As the exhaust pipe portion of them is completely enclosed, you can have holes undetectable by visual inspection.  If they're old and crusty, you'll want to give them a real thorough going over.

The other source for smells is through holes between the engine and passenger compartments.  There should be a fabric sound barrier (looks kinda like a bean bag) jammed between the inner and outer body metal in the upper left/right corners of the engine compartment.  Not only do they provide some sound deadening, they also prevent fumes from drifting up. 

Less likely sources are rust holes, or holes drilled to pass wires or whatnot through.

I'd agree that the heater flaps are first suspect, as if they are closed tight it's hard for fumes to get to the passenger area. Unless... the hoses that go from the heater boxes to the passenger area are missing. If so any fumes under the car can enter the passenger area regardless of all else.

Also, is all the engine tin in place with good seals so that the upper engine area is sealed from the underside where the exhaust is located? The engine fan pulls in air from the upper compartment, and if the tinware seals are missing exhaust can migrate up and mix in. But if the heater flaps close tight and the heater hoses to the body are good, it should still be tight if the heater is off.