Is "exhaustion" of the Hodgkin-Huxley membrane at constant stimulation a real phenomenon?
This phenomenon is called depolarization block and it occurs in real membranes in current-clamp experiments.

The key mechanism is that the membrane has not been allowed to repolarize sufficiently to relieve the inactivation of sodium channels. The Hodgkin-Huxley model reflects this in the "inverted" voltage-dependence of the h gate (sodium inactivation gate)--inactivation is greater at higher voltages. This means that sodium channels cannot open again, or trigger another action potential, until the membrane is repolarized. Canonically, the need to relieve inactivation of sodium channels is the reason for the afterhyperpolarization phase of an action potential where the AP downstroke becomes transiently more negative than the resting potential.

The relevance of depolarization block in in vivo physiological conditions is not well studied. It is not likely that depolarization block occurs in well-behaved neurons under physiological conditions. However, it's possible that it does occur during pathological states.