All dogs need regular exercise, not just being allowed to do their own thing in a space, even a large space, but being taken for walks. Different breeds need differing amounts of exercise. Some breeds need mental stimulation too. (Notably Border Collies.)
The (UK) Kennel Club website has good info on all breeds, including how much exercise, grooming, etc, they need. I think they have a questionaire thingy too, 'What dog would suit me?'
You can keep even a large dog in an apartment, provided you have the time, energy and inclination to give it the exercise it needs. Or choose a breed that doesn't need a great deal of exercise - match its needs to your capability and preferences is the key.
I can't think of any breed which would be both happy and safe (both it and the birds) in 1/2 acre all day alone with hens and geese, whilst simultaneously being a deterrent to a fox. Sorry. Not to mention, as others have said, that the enclosure needs to be dog escape proof and would therefore then be fox proof. If a fox could get in then a dog could get out.
I could be wrong, but I wonder whether your wife's (very understandable) desire to protect the birds is being used as a cover, almost, to get you a dog, to which she's previously been anti. Perhaps it would be better to work on her fear of dogs, so that she is able to accept a family dog, as you would like, and use fencing to protect the birds? I used to have the most lovely natured GSDx, who was wonderful at helping people overcome their fear of dogs. He was such a love. Lots of people commented that they'd been scared of dogs until they met Moses.
