>How [can you] not specify the amount of food/drink(s) that you consume for stuff like burgers...?

#By saying “cheeseburgers.”

The things that don't require articles are **uncountable**, either naturally (water) or because you're talking about the substance (beef) *as* a substance and not as an instance.

There is no ‘cheeseburger substance’ (it's called ‘beef’ or ‘hamburger’) so you **must** use an article. 

However, saying “I eat cheeseburgers” doesn't *necessary* imply that you have ever actually eaten any particular number of cheeseburgers in your life. It's just the way of speaking vaguely that it's something that's acceptable for dinner.

If you really *needed* to underline that you're speaking hypothetically, you'd change the **verb**: “I could eat a cheeseburger” (right now, for this meal) or “I could eat cheeseburgers” (generally, whenever) would both get that idea across. 

**Edit**: Per your comments, “I could eat some fractional amount of a cheeseburger but not possibly a complete cheeseburger” is not an idea actual humans ever need to express (in any language) so there's no easier way to get that idea across that saying it all explicitly.