If you are serious about growing the most calories in the smallest space and in a closed loop sustainable way, you should read John Jeavons book "How to Grow More Vegetables". He has been working on this for 50 years, conducting trials in California and several other countries where food security is critical.
You might also want to read some of the ebooklets they have put out. They go into a lot more detail in the design process in booklet 31. I wouldn't start there. I would start with his regular book, which I mentioned above. I also really recommend the newest booklet 38.
http://growbiointensive.org/ePubs/index.html
In summary Jeavons recommends allotting 30% of diet crop area to high calorie root crops. There are only a handful of them... potatoes, sweet potatoes, Jerusalem artichokes, garlic, leeks, satisfy, parsnips.
Then, in order to grow your own compost, allot 60% of area to cereal grains like corn, sorghum, Amaranth, quinoa, wheat, rye, millet, etc.
The remaining 10% are for veggies. Beans would be considered a part of this area. Also peanuts. Even Cassava would be in this group. It is high calorie, but takes more space than potatoes, for example.
To answer your question in the shortest way possible, the 30% special root crops will produce the most calories per area. However they don't have enough carbon material to create compost. Nor do they make a complete diet. That's the reason for the other 60% and 10% areas. Jeavons has found that the smallest area needed to do all this is between 1000-4000 sf per person.
BTW, I was shocked when I went through all the charts recently and found that not even nut trees can produce as many calories as these root crops on a per square foot basis