What breed of chicken, type of wire, how well is the fence going to be built?
If you get light weight breeds such as the leghorns, minorcas, hamburgs, easter eggers etc a 6 foot high fence will not keep them in.
Most bantams will go over a 6 foot fence without much trouble.
If you get heavier breeds a four foot fence MIGHT stop them. As young birds a 4 foot high fence isn't an obstacle for heavy breeds but as an adult a 6 foot fence usually prevents them from flying over but not always it depends on how determined the bird is.
If just using chicken wire that is not attached to a cross brace at the top I've seen hens learn that all they have to do is fly up about three or four feet and the wire bends outward and they then just walk the rest of the way up and over. Even with heavier fencing if not stretch properly it too can bend outward if a heavy chicken lands on it.
My first run I had was a 8 foot high fence. I used 4 foot high chicken wire and stacked it. I also ran 2x2's at the top and stapled the chicken wire to that. I used hog rings to connect the chicken wire together.
My run was rather large and I had a variety of birds, light and heavy weights and some bantams. I had few birds that flew out but the 8 foot kept 99% of them in. The few that did escape got their wings clipped.
For the record I don't like clipping wings. I don't like how it looks,it does ground the bird but also makes them clumsy on the roost etc.