According to this article from the Seattle Times, potatoes planted inside a box with this method can grow up to 100 pounds of potatoes in just 4 square feet. All that is required:
Careful attention to watering
The Times’ guide for building a potato growing box yields up to a 100 lbs. of potatoes in a mere 4 square feet is shown below:
Plant as early as April or as late as August 1, with an approximated 3 month till harvest turnaround time.
Cut apart larger seed potatoes, making sure there are at least two eyes in each piece you plant.
Dust the cut pieces with fir dust, which seals the open ends from bacteria.
Fertilize with 10-20-20 fertilizer at planting and a couple of times during the season.
Water so that the plants are kept at an even level of moisture.
Don’t plant in the same area in consecutive years or use the same soil to fill your potato box, as potatoes can attract various diseases.
“To save space, Lutovsky recommends building a box and planting inside it, adding sides to the box as the plant grows and filling the new space with mulch or soil. When the plant blossoms, it starts setting potatoes in this added soil. Soon after that, you can start removing the bottom boards from your box and “robbing” the plant, reaching in carefully and pulling out new potatoes.”