I lived for years on a grocery budget of $20-21 per week, and it was not difficult. It was also not terribly unhealthy, though not exactly healthy either. My diet was heavy on carbohydrates and practically meatless. I almost never went out to eat (and got the $2 chili when I went out with friends), and did not buy alcohol. I was never hungry, and actually overate, maintaining an unhealthily high body weight.
In the beginning... there was ramen, and it was bad. Those little packages of ramen noodles are just processed flour, salt, and fat, but they're six for a dollar. I wouldn't eat them alone, though. For the one year that I ate a package of ramen daily, I added a couple turkey-dogs (also predominantly fat and salt) and either three sliced carrots or some frozen broccoli. Actually, it was just carrots for a few months, and I switched to broccoli when my skin turned orange. Broccoli is maybe the second healthiest vegetable in the world, right after spinach, and may be responsible for protecting me from the rest of my diet. Besides the big bowl of ramen'n'at each day, I also ate cookies. Yes, store-brand cookies are a very cheap source of calories, and sometimes some nutrients. A friend recently reminded me that I once figured out how to live only on lemon cookies, but, fortunately, I never tried. I did not have any notable health problems during this year that I recall, but I was young.
The other years were better. I typically had a bowl of cold cereal for breakfast, often corn flakes or frosted flakes with non-fat milk. These were huge bowls, easily three servings or more. I made myself two sandwiches for lunch, using whole grain bread, all natural peanut butter, and strawberry jam. These sandwiches were maybe around 800 calories each. Dinners were usually spaghetti with Prego sauce and broccoli, and when I lived in a place with a stove I often sauteed fresh onions and bell peppers in olive oil to add to the spaghetti. I may have been cheap, but Ragu is just too nasty to eat. My portions of spaghetti were also huge. Depending on sales, I would sometimes buy Tropicana orange juice, bananas, or fancier cereals (Oatmeal Raisin Crisp, Honey Bunches of Oats, Life). I no longer bought cookies or ramen, and I still did not buy meat.
I ate in the neighborhood of 3500 calories each day, and weighed 230-240 lbs. Since then, I have significantly broadened the variety in my diet, and reduced how much I eat, with very good results. I also spend twice as much on food, but because I can and I enjoy it, not because I have to. But I still hold on to the gamer mentality at the store, follow my rules, and try to optimize my gastronomical enjoyment to cost ratio.
A huge problem with eating healthily on a poverty-level budget is not that healthy foods are too expensive, but that unhealthy "foods" (cookies, candy, frozen pizza, etc...) are cheap and easy. Someone has to be knowledgeable and motivated in order to make healthy choices and do some preparation and cooking instead of taking the lazy and immediately gratifying road. It would probably be good for national health to raise the prices of unhealthy foods, and not lower the prices of healthy foods, motivating the highest risk groups to buy better food, especially for their children. Childhood obesity and type II diabetes are serious health and economic issues with some straightforward solutions.