Observability of Orbitals and Orbital Energies
Excluding the trivial case of a one electron system, where the wavefunction for the system is an orbital (1 electron wavefunction) and so could be measured via the electron density, orbitals are not observables. There is no known Hermitian Operator that returns orbitals (or their energies) from its expectation value. So is that the end of the discussion? Not necessarily.  

In a 2006 paper, Shabazian and Zahedi$^1$ argue that what constitutes an observable depends on the conventions we establish for describing a system. They give as an example the Quantum Theory of Atoms in Molecules (QTAIM) developed by Richard Bader$^2$. Strictly speaking, the normal formulation of Quantum Mechanics doesn't allow us to describe subsystems within a molecule. We form molecules from atoms, but once the molecule is formed, there are no longer atoms; rather there is just a single wavefunction which describes the whole system. This is horribly inconvenient from a classical chemists perspective, as much of our understanding of chemistry comes from subdividing molecules into important functional groups that give the major properties of the molecule. To resolve this, Bader proposed a definition of an atom in a molecule by defining boundary conditions for a atom based on flux of the electron density. This gives an unambiguous, experimentally measurable way of determining observables like atomic charge, which without this set of external conditions is not considered an observable at all. While this has as of yet not been done, Shabazian and Zahedi argue that orbitals could be made observables by establishing a similar set of external conditions.   

So while orbitals are not currently observables, what is and isn't an observable can change as new theoretical models develop. 


Shahbazian, S. &amp; Zahedi, M. Found Chem (2006) 8: 37. https://doi.org/10.1007/s10698-005-8247-4
Bader, R. Chem. Rev., 1991, 91 (5), pp 893–928
DOI: 10.1021/cr00005a013