For pygmy goats, a dog house or other small shelter can be about twice the height of the goat measured at the shoulder. As long as you are able to open or move the structure to clean it thoroughly, this small size is fine. For larger goats that must be handled as livestock, you want their shelter to be big enough for you to walk in comfortably and perform tasks such as cleaning, grooming, hoof care, vet care, milking and the like. A standard square or rectangular three sided shed should be at least four feet high at the rear and six or eight feet high at the front.