What you wrote is an expectation value, which means an average on some state $|\psi\rangle$ over all possible eigenvalues of the operator under analysis, weighted with the probability of that eigenvalue occurring on that state $\psi$.
So, yes, $\hat H$ is an observable (we reserve this term for operators and the quantities they represent) and this means that its eigenvalues (the energy levels) can actually be measured by means of suitable experiments.
But $\bar E$ is not an observable, and in general each single result of your measurements might have nothing to do with that value.
How to reconcile the two things?
Actually, repeating the same measurement an incredibly large number $N$ of times (or on an incredibly large number of systems) in the same initial conditions, should provide you with a set of results, whose average, weighted with the frequency each value occurs, should provide a value which bill be closer to $\bar E$ the larger $N$ is (they will coincide in the limit $N\rightarrow\infty$).