For raccoons, opossums and other ground critters 1/2" hardware cloth is pretty good. The main thing is to have something heavy enough that these animals can't bite or pull holes in it, or reach through it with their front paws. In really bad cases, an electric fence wire might be a good addition as well.
Chicken wire is no good for these animals, they will go right through in no time at all.
Don't forget that they can dig under the fencing, at least 12" out from the fence bottom is needed, 18" is what I have found to be the best, that way they don't actually find the outer edge when they decide to dig next to the fence to go under. Just make sure that splayed out fencing is buried a few inches or deeper so they will become discouraged quicker when they have dug easy and strike the wire.
While you can use chicken wire for a topper on a run, if it is just hawks and owls you can get away with just some single wires stretched over the run in a checkerboard pattern. With Raccoons, you would be better to use something heavier and closer knit like the same 1/2" hardware cloth or even 14 ga. 2 x 4 welded fencing wire. Once again it may be that an electric wire or two at the top of the side wall becomes necessary to deter the most determined of predators.