There is something paradoxical about the idea of a stasis with resource depletion, because the pre-depleted stages of the "stasis" must be quite different from the post-depleted stages. Here are some things that would run out quite fast: copper, tin, silver and gold deposits that could be exploited with medieval technology. Recycling of these metals would be feasible, but there would always be losses, which would over time exceed newly smelted replacements. I would think that in late-stasis, non-ferrous metals would be much more rare and valuable than early on. Also, deposits of unusual stone - like the white marble worked by Michelangelo - would become depleted. Also, even primitive humans were responsible for the extinction of many animals, and it's quite believable that a continent like Europe could be entirely without predators like wolves and bears if it got stuck in a long medieval phase.
The fact that even the Romans could achieve significant deforestation and overfishing tells me that this would be a serious concern. This means that any believable stasis would include periods of environmental recovery. In Europe, plagues and barbarian victories allowed some of this recovery, and we might think of other means of periodic massive human dieoffs that allow for environmental resets. Each phase of recovery would be a bit harder than the last, because of the resource depletion.
A less obvious enemy of stasis is the domestication of wild species, which wouldn't backslide in crises, and would produce increasingly useful crops and farm animals. This would gradually increase the efficiency of agriculture and allow for greater population density over time.