Apparently sheep are pretty easy to contain and hair sheep are even easier (no shearing as they shed). But sheep are very sensitive to copper so any mineral you have out would have to be copper-less which might be difficult for you. And it might be difficult to find hair sheep - at least it was for me...though it really depends on where you live.
As far as goats go, in my limited experience, it seems like the smaller the goats are, the more they want to escape.
My two "large" breed goats (well, one is a large breed, the other is "medium") have so far been very easy to contain. The "medium" is an Angora ,who are apparently traditionally very easy to contain. She's about the best goat I have ever met. Snuggly, loveable, follows "the rules", doesn't fuss, etc. She is 9, almost 10, though so I'm not sure how much is breed and how much is age.
The other goat is a LaMancha/Alpine mix and he is a wily creature (mostly, probably, hopefully, due to his age - 11 months old). He's always climbing things, fussing, trying to "help" me in the tack room, etc. He's not "bad" really, he's just into everything. However, he doesn't have any interest in going through fences. He sometimes thinks about it and he has gotten out once but after that time (and after, I think, realizing that his friends were still in) he hasn't gotten out/tried to get out -and he could easily get out- again. Also, his personality pretty much fits with a lot of what I have now read about the general personality of Alpine goats...so perhaps I should have done more research.
The bummers about the Angora is that she needs to be sheared twice a year, she's somewhat of a "hard keeper", and she needs to be kept dry when it rains (she wears blankets). Otherwise, she is a really perfect goat. And really, those things are not that hard to deal with - I just feed her extra feed (the goats have a separate "goat area" that Lacey can't access), it's not hard to shear her with scissors or clippers, and I blanket Lacey when it rains anyway so I just blanket them at the same time.
I would love to have SO MANY MORE Angoras one day!
I have heard that cows can be hard on the ground (aka - MUDDY) and they can really lean on fences. However, I have no experience with cows so... haha
Llamas, Lacey used to live with a pair of llamas and they were great. They were semi-wild so that was sort of a downer (they would sometimes get in the way but there was no way to really contain them without them freaking out) but otherwise they were very easy. They didn't eat a whole lot and they were pretty social/not mean.
They did/do have a very highly developed predator sense so it was sketchy trying to have a dog around them but I'm sure that a dog they saw everyday would be fine ("my" dog had a pretty keen prey drive so, even though he was totally under control, the llamas did not particularly like the way he looked at them ).