Consider a project like the Earthship. Supposing one would like to grow food (organic polyculture) indoors in such a habitat (i.e. in a green house). How many m² would be required to feed one person yearly?
update Answer has been edited after considering comments by bstpierre.
This is a great question, and I hope to learn more from some of the other answers and comments. I was unable to find good information that can be used to answer this question, but based on a few informed estimates, all assuming a vegetarian diet.
0.04 ha / 0.1 acre
John Jeavons (see answer by bstpierre):
0.01 ha / 0.025 acre
Eliot Coleman Perhaps a minimum estimate, also see comments
In Elliot Coleman's book The New Organic Grower, the author claims that he can grow enough food for 100 people on 2.5 acres, equivalent to 0.025 acres / person, 1100 ft2 or 100 m2. Elsewhere, he claims to feed his household of two adults with a 40'x40' plot of land plus a 20'x40' greenhouse (1200 ft2).
It is interesting that the author has 25 years of experience (at time of publication, 1995) cultivating 5 acres of land in Maine intensively using an eight year-rotation.
0.07 ha / 0.17 acre
The minimum amount of arable land required to sustainably support one person is 0.07 of a hectare. This assumes a largely vegetarian diet, no farmland degradation or water shortages, virtually no post- harvest waste, and farmers who know precisely when and how to plant, fertilize, irrigate, etc.
0.02 ha / 0.05 acre
A minimum estimate of the amount of land required to feed a crew on mars, but the only estimate here that is 100% indoors.
These estimates are minimum requirements, assume a vegetarian diet, and depend on highly skilled farming.
I've seen from multiple sources that about 4000 sq ft per adult is about what is required, assuming a vegetarian diet.
Jeavons' book goes into some detail about how to plan your garden and your diet. Around p25 there are charts and explanation about choosing crops that are "Area- and Weight-Efficient". E.g. potatoes, leeks, garlic, parsnips. According to the plan in the book, 60% of the planted area will be high-carbon producing and calorie-producing (grains, sunflowers, nuts, grapes), 30% will be hi-cal, area-efficient (potatoes), and 10% will be "vegetables" (broccoli, lettuce, etc). Under this plan, a large portion of your calories are coming from food like potatoes.
Reading the few pages from Jeavons' book points out how intertwined the planning of both your diet and your crops must be. If you plant 4000 sq ft of corn, beans, and carrots but neglect to provide enough calories through potatoes, you may not have enough to eat.
Also critically important, but not covered much in either book, is the importance of being able to store your harvest past the growing season. Again, even if you do grow a year's worth of potatoes, you'll end up starving if everything is spoiled by the time January rolls around...
Update: a table in "The Permaculture Handbook" (Peter Bane) on p109 provides a breakdown for ~2700 calories per person per day (~1 million calories per person per year). It's not a vegan diet: the list includes milk, cheese, meat, eggs, and fish. It's based on the premise of a closed-loop fertility system so that manure, compost, etc is created on-farm instead of being imported. Bane's land figure comes to 14,500 sq ft (1/3 acre, 1350 m2).
An issue that Bane mentions is also worth considering: what are your labor and energy inputs? You can get more output from a given area of land by putting more labor into it. You can decrease labor by using energy, but if you're supposing a closed-loop system then you need more land to grow energy crops. Importing energy into your system doesn't eliminate the need for land to produce energy, it merely moves it. (And if your energy is from petroleum, you still haven't removed the need for land, you've only moved it back in time a couple of million years.)
This is not a precise answer to your question, but Lolo Houbein in 'One Magic Square' claims that 1 sq metre (9 sq ft) can provide 1 salad meal per day for two adults (or smaller side salad for three) all year round, through the application of companion planting techniques.
If you extrapolate this idea (and eating only salad three times a day) 3-4 sq metres would be enough to completely supply the diet of two adults. Of course, stuff like pumpkin and watermelon cover huge areas so if you had only this small a space certain things would be excluded from your diet.
So, even though it's not a precise answer I hope it could be helpful for others reading the question, who are trying to grow all their salad/vegetable needs in one backyard, not necessarily grow enough to cover their entire diet.
The family in the following video has 4 members and they use 1/10 (0.1) acre to feed themselves and get $20.000 a year by selling surplus food. If you divide that by 4 then you get 0.25 acre per person, similar as the other answers. Their diet mostly includes plenty of vegetables, chicken, eggs, honey and some grains.