What's the average amount of essential nutrients that I absolutely must be eating in order to not end up in the hospital?
To answer this specifically about what it takes to prevent you from dying of malnutrition -- the human body requires enough calories to cover energy expenditure, sufficient intake of all 8 amino acids that the adult human body cannot itself produce and sufficient micro-nutrients to prevent diseases such as scurvy and so on.
The average young man who doesn't exercise besides some light walking will burn about 2000 - 2200 calories a day. This can, for the most part, be any fat or carbohydrate. The body can process these about equally.
Most grains and vegetables do not, by themselves, form a complete protein -- containing adequate amounts of all 8 proteins. Meat that has not been heavily processed is a complete protein. Unprocessed grains and beans form a complete protein as well. Corn is seriously lacking in a couple of amino acids and you cannot survive on corn alone. The different sources of protein don't have to be eaten at the same time... variety over the course of a couple days is sufficient.
As for micronutrients, an assortment of fresh foods will normally cover the requirements. If the diet is going to be monotonous (rice and beans every meal for more than a couple weeks), a multivitamin supplement will usually be sufficient.
A total starvation diet (no food, just water) will usually lead to death in a healthy adult of average weight in 2 to 3 months.
A diet sufficient in micronutrients and amino acids but lacking in calories will result in gradual weight loss, first of fat, then muscle mass, ultimately leading to organ damage if continued indefinitely. The rate of loss is corresponding to how much of a deficit between food intake and expenditure is.
A diet lacking in micronutrients or amino acids with or without sufficient calories will result in a variety of nutritional disorders, most of them rather unpleasant. Scurvy, for example results from vitamin C deficiency and would set in within a few months of eating a solely corn or rice-based diet.
There's a lot of good posts above that show you how to meet your minimum requirements for as little as $1 - $2 dollars a day. Combinations of rice/corn/potatoes and various beans and vegetables can be very cheap and filling, they can even be quite tasty. Potato/bean stews can be very cheap and fairly simple to make. If you have any more money, very cheap cuts of meat can be cut up small and cooked on low in the stew for 6+ hours to soften it up.
Avoid eating the same thing every day because it's possible an essential nutrient is missing from that particular combination. A daily multivitamin and a single can of salmon a week will eliminate any concern of nutrition deficits, in which case you can concentrate on covering your daily caloric need.
Good luck.