Did Athens and Sparta suffer from lack of food?

It depends on what stage of their Spartan life you are referring to.

I have read that during their early years, during military training, that spartan boys were sometimes fed pig fat and pig blood. The idea was that if you could stomach anything you would have a better chance of surviving. It also encouraged the boys/trainees to steal food. When caught they were punished not for stealing but for being caught. Again it was meant to ensure their survival.

Once they graduated or became men I believe they could eat anything, but their diet was still somewhat limited. Because Sparta discouraged trading there would have been little exotic foods. The diet would of consisted of whatever the Helots, the Spartan agricultural slaves, grew. Which would have been whatever was native to southern Greece.

Sorry, I can't be more specific.