I think a sustainable choice would be to start producing my own food in my backyard. To achieve year round production, I'll need a greenhouse. We enjoy lots of vegetables and salads on a daily basis, and will be using this as a supplement to our animal based protein diet (so it won't be the only source of food).
What size greenhouse should I expect to need in order to feed a family of 4?
What can I expect as far as potential yield per sq/ft?
The answer is: "it depends". In theory, you could use a number of different models (e.g. the infographic in Daniel Bingham's answer, John Jeavon's "How to Grow More Vegetables", etc). You'll get a wide variety of answers, and a lot depends on your specific circumstances -- and will likely vary widely for you from year to year.
Note that a limiting factor may not be so much the cold, but the fact that the plants get fewer hours of sunlight in the winter as you move to higher latitudes, so they're not growing as much in the winter. Eliot Coleman talks about this in some of his books. One strategy you can use if this is an issue is to start your plants that will be harvested during the winter in the late summer so that they can have enough time to grow before the short days of winter arrive; they will be dormant (i.e. not growing) at that point but still harvestable.
I think an appropriate strategy would be to start small and experiment to figure out what your right size is. Set up some hoops over your garden to protect a handful of different winter crops. Keep track of how much space you allocate, what you've planted, and how much you can harvest and eat. Use this information to scale up your winter protection, and possibly to make adjustments to your crop selection, management, and harvesting.
A complementary strategy would be to increase the size of your summer garden so that you can have vegetables preserved for winter use. Winter squash, potatoes, and onions all store easily, and many more vegetables can be stored for months if you can provide a proper storage facility. (Apples and other fruits too, if you take a food forest approach to gardening.) Even if you don't have a good root cellar, you can preserve many other foods in a variety of ways. If you're processing your own chickens when there's a lot of vegetables available, use them in vegetable-heavy soups. (It makes the meat stretch farther.) These soups can be canned and enjoyed all winter long. I keep bags of greens (kale, spinach, chard) in the freezer for use in recipes during the winter -- lasagna, soups, quiche, etc. If you don't like the energy used by a freezer, most of these can be dried effectively as well, and stored for long periods that way.
Finally, to put a number on it, let's say that you want to provide four 4oz salads (4oz of fresh greens only) a day for 6 months (October 1 through March 31). That's a pound a day for 180 days -- 180 pounds. A decent yield for spring-grown spinach would be about 1 pound per square foot, lettuce perhaps 2 lb/sqft. To make up a number, let's say you average about .5 lb/sqft for winter-harvested greens. You'd need a greenhouse with 360 square feet of growing space, plus space for paths, whatever heating you are going to provide, and whatever else you might keep in the building. It's a fairly simple formula, so you can play with the numbers, but until you know what your numbers really are (i.e. the strategy above), it's just a guess.
(I'm assuming greens above because they're among the easiest things for winter harvest, and to keep the answer simpler. Carrots and parsnips can be harvested in winter too as long as you keep the ground from freezing around them, but they don't need a greenhouse. I cribbed the yield numbers from Jeavons, because I had it handy.)
This probably depends on exactly how long your winter is. The typical area I've seen quoted to fully sustain a family of four is about two acres.
This is the space required to be nearly self sufficient, including electricity, animals, and grains. But most of that space is taken up by the veggies. Of course, that is to supply the veggies for a whole year.
To supply the veggies for just the winter months depends on how long your winter is. The 7 month long upstate New York winter I use to have to deal with is a totally different ball game than the 3-4 month southern Indiana winter I have now. If we assume a decent winter of about 4 months and extrapolate from the graphic below, you'd need about 25,000 sq ft of green house to feed a family of four veggies through the winter. That number probably varies depending on what, exactly you plant and how you make use of vertical space. Also, how much veggies you eat (I think that number assumes a completely vegetarian diet).
Either way, in order to heat enough space, your best bet is probably to use row covers of heavy transparent plastic over your normal garden, rather than using a green house. Lay in a fall crop of cold tolerant plants, and then build the row covers over the top of them. Row covers are comparatively cheap to build and maintain. They don't have to be anything more than hoops made of small diameter pvc pipe or steel re-bar with transparent plastic over them. If you're somewhere with heavy snow, re-bar is probably the better choice. As is a heavy plastic. If your winter is light, you can probably get away with the pvc and a lighter weight plastic. You can bury the edges a little, or cover them with mulch and stones to prevent them from blowing away in the wind. Again, take into consideration just how rough the weather is likely to get.
Another alternative is to use cold frames, wooden boxes with glass tops generally facing south. These are generally harder to get a lot of area with than row covers, but they are very cheap and quick to build and pretty effective.
Here's a semi decent run down of the types of row covers:
http://www.organicgardening.com/learn-and-grow/row-covers?page=0,0
Here's the info graphic that 2 acres number comes from:
So the 2 acre thing - well - it's a stake in the ground. If you are using traditional methods...I don't think it's actually very far off. I also agree with a previous comment/assumption - where's the food coming for the animals...I think they are assuming this is this a "closed system" - the holy grail of all gardeners - true food independence and begs the question - is there a self-sustaining model available to most people. Ideas like this will absolutely make a difference - where our food is becoming more and more a blind item at markets.
All things can be improved. Vertical gardening techniques certainly cut down on space requirements in theory. Aquaponics...Green houses...Climate...Design plans (Annualized solar), soldier fly larvae for chicken feed (again location dependent) - There's a lot of information out there.
It's like asking a stranger how much they spend on heat a year...and then gasping at the answer...without asking how big of a house they are heating...where they live...the age of the house...if it's insulated properly...what type of heat to they use or what type of heating system.
This really isn't a simple problem. - especially if you are looking for sustainability and a closed loop system/factor.
It requires a lot of process and planning.
I think everyone wants to be independent with all things...electric...heat...rent/mortgage...and food...
The feasibility/effort scale - I'm not sure if most people have that much ambition and foresight available to them. They may not realize that such a project - is a commitment to a certain lifestyle - in order to be remotely sustainable. Permaculture...fascinating!
So processes and systems needs to be developed for many climates with goals in mind for both sustainability and abundance. There is a cost to set up such lofty goals, but I think it's a worthwhile problem to solve and once designed - truly designed - hopefully it will be simple enough, attractive enough and modular enough for others to easily follow.
Way too much depends on your circumstances.
Start with a small one -- 15 x 20 or 24 x 40. In general larger greenhouses are easier to keep at even temperatures. If you're not sure, get the wider one to start but only buy half the frames for it, then extend it as you can.
Heating them can be a step backward in terms of global sustainability. Solar greenhoouses are hard to do, and the more extreme your climate the harder it is. Most greenhouses are used only 3 seasons of the year. (In southern regions, you can't cool them enough.)
Greenhouse frames themselves are quite inexpensive, and you can make them moreso if you bend your own tubing. Poly covered greenhouses require new poly about every 3-5 years, depending on the quality you get. Poly is pretty much the standard for most production greenhouses.
If you have the land, look at buying a big frame -- say 40 x 100, and putting a mobile home inside it. Because most of your heat is from the greenhouse, you don't care if it's an old ratty one with crap insulation. YOu do need roll up sides on your greenhouse to keep from getting cooked in summer.
If you don't plant all of it, the kids have a place outside on a rainy day.
