I've spent a lot of time around goats here in Colo and a little bit of time in Alaska. Most goats will actually allow you to get pretty darn close...especially in the off season! The off season is obviously the best time to get up close and personal with goats. I often try to get in as close as possible without letting them see me. They have fantastic eyes so that can often be challenging! Once they see me....which usually happens I just sit in the same spot for 15 to 20 minutes and let them get used to me. Once they figure out that I'm not aggressively trying to get close to them I'll slowly move at an angle towards them....never looking them in the eye or making any quick movements. I'll move around 10 yards at a time. Depending upon how nervous they are I can move longer/shorter distances and wait longer periods between movements. Obviously this doesn't work every time but works pretty regularly on goats during the off season.
This same thing works on sheep....but with sheep I ALWAYS do this from below. If you stalk sheep from above and they see you they will likely run! Goats it doesn't seem to matter as much if you are below or above. Both goats and sheep are on higher alert during the day.....and often super calm close to sunrise and sunset. Mature billies tend to hang out in the biggest cliffs in the area during the day.....and often come out of the cliffs to feed in early morning and evening.
I put together a long article about goats, goat behavior, etc. Most of the info is based on my experiences watching and hunting goats in Colorado and Alaska. If you send me a PM I would be glad to send it to you.