I've raised dairy goats for over 40 years and have yet to be self sufficient doing it, though I have tried. Most of the dairy goats are bred to eat a considerable amount of grain ration and a lot of
hay in addition to pastures to maintain their body weight and produce
milk. Some Nubians are dual purpose (the old style) and forage pretty well and maintain weight, and the meat type breeds of
course do. They've developed
cattle who can forage only (with addition of hay dry matter as forage has a lot of
water in it, causing bloat in absence of hay when it is wet or overly moist, as well as limiting capacity of what they can eat due to high water content filling stomach), who give milk without added grain input. Goats do well on oats/sunflower seed with loose minerals available at all times, and maybe a sprinkle of
flax and dried molasses, as their grain ration. They do need high forage, not much grain as their stomachs cannot take much grain, it interferes with rumen functioning. I have one doe who cannot have more than two cups of grain ration per day or she loses rumen function, goes off feed, so I have to limit her. The others can take up to a quart at the time. My pastures have a mix of many high protein forbes such as varieties of clover, lespedeza, chicory, and wild and domestic dandelion, plantain. Mulberry
trees hang over the fences and when leaves drop in fall they gobble them up. They are a complete protein and high in calcium. In winter (I'm in lower N.C. mountains), I overplant with winter wheat, ryegrass and sometimes oats. In a hard winter, the oats winter kill but the wheat can take it. They paw down even under snow to find green matter.
So the short answer is, yes, it can be done, but ultimately you will have to really work at it to get good variety and high mineralization on your land to make the good stuff grow. The more variety in the mix the better and goats utilize protein and nutrients in many plants that a cow cannot eat or digest. Just watch out for the poisonous stuff and get a list from your county agent. Black walnut, poisonsous to horses, is ok for goats and it helps keep them wormed. Speaking of worming, rotating your pastures and planting lespedeza, having paulownia trees so they can eat the leaves, the black walnut leaves and many other things can help worm them. Having them on the same land for more than 2 weeks at the time is asking for trouble. Move them around every 2 weeks to break up the parasite cycle. They also love
locust and acorns. Too many acorns can lodge in their rumen, blocking it from movement and can ultimately kill them but it takes a lot. The acorns make them gain weight (it is high in protein and good omega oils), and have higher butterfat and wonderful tasting milk. I sometimes pick up acorns from my trees which have really big, mild ones, and save for late winter as a treat for them and give them a few every day at milking time.
You may also have to do some serious culling and selection to get the animals you want who thrive under the conditions they will be in at your farm.
If you want to work this hard, you could use a scythe and cut your own hay, cure in the field and store loose in the barn loft. I used to do this and had a trap door in the loft so I could open and drop it down into their hay rack and not have to carry it. I would just rake over a portion daily. It is easy to grow oats and sunflower seed, both only take 45 days to mature for harvest.
They eat a lot more than most people realize and unless your pastures have high mineral and nitrogen levels, you will need a lot of land for them. You can't go by the books, because there is poor pasture, good pasture and really good pasture. Lot of difference in what it can accomplish. I use ag lime, soft rock phosphate and
wood ash on my pastures, and lately have been mixing in a little kelp, azomite, boron and selenium because east coast soils are deficient in the last two. Those trace minerals are important and you'll see the difference in the health of the animals, weight gain, milk production and healthy, vigorous babies. We are what we eat! And they are too.