You don't need a community effort or astonishingly vast land to grow potatoes :-) What I mean by that is that the choice of stuff you grow matters. Lettuce, peppers (and one might add tomatoes) won't feed you much indeed. They are however indeed interesting in small spaces when you don't expect them to feed you but to provide you nice, fresh extras.
I mean, I produce about 800 lbs of vegetables by spending 20 mn a day on it (average on 365 days, which means more at times and nothing at other times). Surely it requires more space that you had. But no motorised tool involved, no fertiliser but a tiny bit of manure (no fancy permaculture tricks either, just traditional beds), no pesticide except in case of emergency like once a year on 10% of the garden, no watering except in case of emergency again, no search of any optimisation (time, space, yield, ...). It isn't a bid deal to get a partial yet significant autonomy; it just gets harder and harder as you want to get close to 100%.
There are stuff you can keep across winter in storage without transformation, like potatoes or cereals (onions, shallot don't do bad either); and stuff that can be kept where they lie, in the ground, like parsnip, sunchoke, and a few other root vegetables; cabbage can stay too, leeks as well. (Of course, it depends on the geographical location.)
Yeah, a base of potatoes + cabbages + onions get you a long way; and they are quite versatile as far as cooking is concerned.