The answer is that they do not occupy the same state. The state would be defined by them having exactly the same quantum numbers but trivially one of the quantum numbers for these two particles is different: Charge.
Another way to see thins is two phrase the pauli exclusion principle:
Two identical Fermions cannot occupy the same quantum state at the same time.
But here this is not the case, by for example applying an electric field you can easily differentiate between the two fermions and so they are not identical so the pauli exclusion principle does not hold.
On a separate note you might have gotten the electron falling into the nucleus argument mixed up with something. As far as I am aware there is no link between that and the pauli exclusion principle at all as the particles making up the core are definitely not identical to electrons. Instead the normal argument for this is given by the Heisenberg uncertainty principle, telling you that you can not measure both the momentum and position of a particle at the same time with arbitrary precission. As such if the electron where to fall into the core it's position would be "measured" very precisely and thus the uncertainty in momentum explodes, such that the electron "flies" out of the core. The stable orbit of the electron is where this effect exactly balances the electromagnetic force between core and electron. Now note that this is wildly simplified as Quantmmechanics have all sorts of subtelties but this still gives an intuitive explanation.