The USDA recommendation is just to provide the small amount of vegetables a family normally eats, not to provide all their calories. 50 sq m is absolutely tiny, and even with the most calorically dense crops like a 300 bushel/acre corn crop, that 50 sq m is producing less than 1000 calories a day, it won't support anyone.
As it stands US Department of Agriculture says that 120 sq m is enough for a family of 3 - thats using conventional gardening techniques.