I’m imagining rural or small town eating places, or maybe some joint in the “working class” area of the city. For instance in Michener’s “Chesapeake,” a hotel in Baltimore in the late 1800s had a buffet canteen tucked at the back where people can eat all they want, if they buy one mug of beer. Do you still have places like this, or something similar?
Of course. With a can’t-fail business model like that, how could they not flourish?
Generally, any place that makes much of its money off something other than food can afford to offer very cheap eats as a loss leader, to bring in the customers. Almost-free food used to be common in Las Vegas; I don’t know if it is any more. Cracker Barrel, which has a store attached to every restaurant, is quite cheap.
Cici’s Pizza. All you can eat for about $6 if you drink water; tea or soda add another two.
How cheap is “very cheap”? All-you-can-eat buffets less than $10 are not at all uncommon in some places. Less than $5 would be sketchier but I’m sure they exist.
This may be not what you’re looking for, but I find many Chinese restaurants serve large portions rather inexpensively. I always get 3-4 meals out of my lemon chicken.
Go to the downtown diner in any small town (population less than 5000) in Minesota at noon for their blue-plate special (or “commercial” in some places). You’ll get a heaping plate of mashed potatoes, thick slices of ham or roast beef carved by hand from their own roast (not from Sysco), a gallon of brown gravy, one or two cups of canned corn drenched in butter, and a slice or two of thick white bread. Usually $4 to $7, maybe $8 if they’ve redecorated the place this century.
Then have a huge slice of apple pie for dessert. Another $2.00.
Then go back to work and have a nap.
That reminds me, some Harris Teeter deli departments sell huge paninis for about $6. I can get at least two meals out of those.
Home Café, New Carlisle, Indiana - huge buffet, all you can eat, all items homemade. Nothing gourmet here, but if you like scratch made mac and cheese, homemade noodles, pies made on the premises, corn grown a half mile away, etc., you’d love this place. I haven’t had the pleasure of eating there since I left Indiana in 2002, but at that time, you could still get your dinner for less than $10 and the lunch buffet for less than $7
Sunnyside Grill in Cleves, OH (west side of Cincinnati). The standard hamburger is 3/4 lb. They have double decker sandwiches that are 6" high. Last time I was there it cost me $8.
Similar in New Jersey. The landscape is littered with diners, and most serve decent food in large portions for reasonable prices.
The Hot Dog Shoppe, East Liverpool, Ohio. Great hot dogs, fries and shakes, and very inexpensive: http://www.yelp.com/biz/hot-dog-shoppe-east-liverpool
Fall, 1977. Fiasco Restaurant, Marina del Rey. 4 - 6 p.m. Monday through Friday. Happy hour in the lounge. Buffet line over next to the wall has hot and cold hors d’oeuvres, cold cuts, cheeses, breads, potato and macaroni salads.
You buy a cocktail in the lounge ($3 - $4), you’re invited to help yourself. I’ve seen similar happy hour spreads at other mid-range restaurants, but not very often, at least since the 90s (I’m not often going to restaurant lounges at happy hour, though).
Nancy’s Home Cooking in Columbus might fit the bill; more of a college dive than a working class eatery, you pay somewhere south of $4 and get a gigantic plate of whatever they made that day. I only went twice, it was chicken and noodles both times. I heard they had to raise their prices at some point but who the hell knows how they managed to keep the lights on charging $3.65 for a sit down meal.
That was the first place I thought of. They’ve gone up in price; back about 15 years ago, they were $2.99.
Testify!! Went out east last September and ate at this place. I was so stuffed I, unfortunately, had to leave most of the meal behind. Good thing I had walked from the motel so I could burn a little off.
City Cafe in Northport, Alabama. I used to get a 3 or 4 vegetable plate with cornbread and tea for about three dollars. It was all delicious.
https://www.zomato.com/tuscaloosa-al/city-cafe-northport/menu#tabtop
Another (possibly apocryphal tale from the 1980s) has the San Jose IBM site cafeteria listed in a “cheap guide to visiting California”, since it was subsidized by the company as an employee benefit. So much so that they had several tourist buses stop and unload hordes of senior citizens who got in line with the engineers and manufacturing workers to grab some cheap food. It didn’t take long for them to put a stop to that.
I don’t know if you can stop at any company cafeterias any longer, or even college cafeterias, to get a cheap place to eat. I assume there are security concerns at most of those places. Does anyone know of such places?
Oh, that’s funny. I used to live near LaPorte and was just raving to my husband a few weeks ago about Miller’s Home Cafe. When I was a kid, we’d go a couple times a month and I would pig out on chicken and noodles and their stuffed crab shell things. 
We’re going to go by there next time we road trip to Chicago.
In my experience, college cafeterias are open to the public, but serve low-quality food at high prices. Unless you are visiting a student who has guest passes to the cafeteria, don’t bother.