The phrase "running on fumes" literally denotes that one is almost out of gasoline, ostensibly in a vehicle with an internal combustion engine. This saying is built on the fact that gasoline is known to evaporate into vapor. These fumes are quite flammable. This is often also referred to as "running on vapor(s)", "running on empty", or "running on E".
This saying is used metaphorically to describe any situation where you are running low on any critical resource, whether it resembles fuel or not, though especially when it does; it is commonly used to describe a state of hunger.
Similar to running low, especially in usage. Also similar to running on reserve, though that typically indicates using reserve power or a reserve (fuel) tank. Opposite of topped off, or (more simply) full.