When my kiddos were really young and we as a family of 6 were living close to or below the poverty line, I was feeding our entire family on roughly $120 a month.
It takes a lot of discretion and a lot of planning, but I'd be able to go to Aldi's down the street and stock up.
Of course, everything was cheap. And I'd look in other carts where families would be purchasing their food with the SNAP cards (we never went on public assistance), and the choices were all frozen convenience meals that were single-serving. We were able to stretch a whole roast chicken for a week. Still do many times (though now it's the roast chicken from the local farm).
Bags of rice, cheap eggs, limit the cereals to only toasted o's, grate your own cheese, learn to bake from scratch (that's how we got our treats from flour, sugar, and blocks of unsalted butter we'd get on sale and freeze). And of course, frozen veggies, salad veggies, and heavy cream....the last ingredient helps to make ice cream and thickens creamy soups and stews for end-of-the-week meals. We never bought cheap sodas or soft drinks there, either, or packaged chips or cookies.
If husband was adamant about not going onto public assistance for whatever reason, and if he trusted me to make a miracle out of very little, I found a way to do that. I baked all of our breads, made our sauces and salad dressings from scratch, and whipped our own butter and added herbs from our growing garden (I was still a newbie back then).
Heck, I even found a way to dehydrate beef jerky in our oven for snacks.
It's entirely possible to live on that kind of food budget. The only problem I have with that kind of eating is that we wind up spending our money toward food companies with ethically questionable or objectionable behavior with their commodities, with the animals they raise and how they are killed and processed, and their staffing of undocumented migrant workers. Plus, long-term eating habits that include very cheap carbs and meats that are cheaply corn-fed tend to result in various health risk factors when cheap carbs and meats aren't tempered with quality fresh fruits and vegetables.
If we had to go that route again, our family would be able to adapt. If we needed public assistance ever in our lives, I would have little doubt to our ability to feed ourselves adequately.
My only concern is the sustainability of the various farming practices in order to provide the bulk of all these cheap foods.