I'm going to be honest: there is absolutely no way I can achieve the amount of calories necessary to make a dent in our family's total yearly intake without growing some seriously calorie-dense food. So, to be Socratic about the whole thing... What food crop can offer high calorie yield in a small space?
Genetically modified Monsanto corn is grown at about 200 bushels per acre. Wheat yields are about 100 bushels per acre with all things working to plan. Potatoes range from 300 to 400 bushels per acre!
Another point to make is that not only do potatoes yield higher per acre, but potatoes are also far more calorie dense than corn, wheat, rice, and soybeans. The potato Wikipedia entry claims that 9.3 million calories per acre are possible from potatoes. Clearly, it seems that the calorie density of potatoes per acre makes it the most valuable choice for this project.
So, given that potatoes appear to be the best choice, what is going to be the best way to grow them? Clearly, an option is to just sew seed potatoes into the raised beds and let them go. The problem with just sewing into the garden bed lies in the difficulty in continually mounding dirt up on the potatoes without blocking the sun for other plants within the bed.
Youtube has a bunch of videos available with techniques that seem to come from fresh ideas about sustainable agriculture and how to maximize space in the most efficient way possible. I've had my interest piqued with the above-ground method that makes use of bags of various types.
I've decided to take it all one step further. Landscaping cloth is a synthetic rolled material that people typically use underneath flower beds to keep weeds from coming up. I'm going to make bottomless framed bags more like the kind in the second youtube video link. I plan to use 1x1" lumber to stake out the corners of the bag and then staple the landscaping cloth to each post. As the plants grow up, I can add soil and compost and roll the landscaping cloth up, stapling as needed to keep things tidy.
The whole point is that, as far as my reading has found, potatoes will grow along a plant stem wherever it is covered by soil. So, it'll be interesting to see how far up I can push this method and what effects it will have with the potatoes. As far as I can tell from all the youtube videos out there, this method is far from fool-proof.