The last three apartments I've lived in I've been able to have a garden. Not a large one mind you, and I've had to learn how to utilize grow bags rather than growing directly in the soil, however home ownership isn't a requirement to being able to grow your own food.
The last apartment complex I lived in, every apartment had their own ground rear patio, with about 10ft of grass from their patio to the wooden dividers giving you a bit of privacy from neighbors.
One family across the way from me had their patio fenced in with chicken wire (with an access door for maintenance people) and the main grass area had been redone for gardening, and planter boxes put against the walls. Their garden faced south so they got plenty of sun, and the few times I was on that side of the apartment property and got a look, they had quite the garden overflowing.
No idea how they got management to agree to it, but it was there for at least 3 years before I moved.