Consider the following quantum system: a particle in a one dimensional box (= infinite potential well).
The energy eigenstates wave functions all vanish outside the box. But the position eigenstates wave functions don't all vanish outside the box. Each one of them is a delta function at a specific location, and some of these locations are outside the box.
So it seems that there is no overlap between certain position eigenstates and all energy eigenstates. So the energy eigenstates don't span the whole Hilbert space! And these position states have zero probability for any energy outcome in a measurement!

Now, I know that when speaking about an infinite potential well, it is assumed the particle cannot be outside the well. But I don't see any reason to assume this from the postulates of quantum mechanics. Is there an implicit additional postulates that says: "The Hilbert space of the system is spanned by the Hamiltonian operator's eigenstates (and not other operators, such as position)"? Or is the infinite potential well just an ill defined system because it contains infinities (just like free particle...)?