Not in our heads, but in our nervous system. Some types of fatigue are due to feedback into our central nervous system that reduces muscle activation signals from your brain in order to avoid damaging the muscle. But if you're untrained, your body doesn't actually know where this point really is.
Most strength and endurance gains when you first start training are neurological, where the effect of this feedback loop is pushed back as your body learns the true threshold for muscular damage.
The heart does get fatigued though. You can run a horse into dying from a heart attack, for instance. Most people simply aren't fit enough to push it that far, as their other muscles would give out first.
It's the same with our body. Feeling tired is separate from being tired. Feeling hungry is separate from being hungry.
Drugs work like putting electrical tape over a low oil light. Amphetamines and even caffeine are good at blocking out the signal. The underlying issue is still there.
But, sort of like you said, the signal comes really early. It's like a low fuel light that comes on when you get to a half tank.
That's only true for some drugs, those that mask pain. Those that stimulate the mind are like feeding an engine nitrous...and sometimes similarly dangerous!
That’s a far cry from “all in your head”. It’s a systemic response that affects every part of the body.
Military schools like the Ranger Indoctrination Program really emphasize this point. The body will go on far past the point you think it will stop.