In all gardening, start with your life. What do you like to eat? How much time can you devote to gardening, really? I always suggest that people start small, something between 32 and 150 square feet. In really tight spaces, you may have even less garden. Not to worry, the most challenging gardening is the first 10%.
The philosophy of square foot gardening is to maximize the value of small spaces. Square foot gardening is a very old idea. F.H. King wrote about small space intensive growing in his 1911 book, Farmers of Forty Centuries, about his travels through China, Japan and Korea. The farmers of those countries were feeding themselves and producing a surplus for sale on postage stamp sized plots. The market gardeners outside 19th century Paris fed the whole region from tiny, intensively managed farms. Mel Bartholmew, a laid off engineer, popularized the name square foot gardening in his book and PBS show.
The basics are no different from organic farming: feed the soil so that it feeds you, rotate crops, use cover crops, compost, plant in raised beds or wide rows. Small space gardening has some special quirks:
1. Use vertical space. Trellis everything that you can to minimize the footprint, including squashes and cucumbers.
2. Space plants using their "in row" spacing. On the back of an envelope of carrots, it says something like space 3" apart in row with rows 30" apart. You are going to plant a solid block of carrots spaced 3" apart each way, for a total carrot density of 16 carrots per square foot.
3. Plant continuously. Plant something every week until you finish the year with a little garlic patch in October. Growing your own starts will keep the cost down. Stagger plantings to give yourself a continuous harvest.
4. Plant when you harvest. As you go out and pull up a head of lettuce for dinner, plant something in the spot: a cabbage start, a few green onions, some spinach seed. I'm casual about this one, myself, as I am more inclined to harvest dinner and run back in the house to cook.
5. Your basic unit is the square foot rather than the row. A square foot holds 36 green onions, 16 carrots, 4 loose leaf lettuces or bush beans, one cabbage or broccoli, 36 radishes, etc. A tomato plant requires 4 square feet in a cage or 2 square feet if trellised. A square foot garden looks like a patchwork of different vegetables.
6. Plan for season extensions. Row covers will warm up the soil and let you plant your warm season crops in May, when the weather is generally not settled yet. A hot box will grow lettuce into December.
Some thought on the potential of a 4' by 8' raised bed reveals that you can grow a great deal of food in small spaces.
Searching youtube for "square foot gardening" will turn up 14 gazillion results and you'll be an expert in short order. For a great sophisticated low tech gardening solution, search youtube for the Lesotho keyhole garden.
By the way, F.H. King's book is available by interlibrary loan, or as a free download without the pictures. This last is of dubious utility, since the whole book is essentially a photo essay. If someone is feeling motivated to raise the funds to purchase a copy of the 1911 hardcover edition, it would make a useful addition to the library system here.