Mary is correct, there are trade-offs. The more predator proof you want to make it the more you need to spend. Any mesh fencing will provide some protection. The heavier the gauge the more likely it is to stop something big. Larger dogs, big male raccoons, and some other big critters can rip chicken wire or even hardware cloth. 2x4 welded wire will stop big critters up to a bear. Electricity is about the only thing that stops a determined bear. But the larger the mesh the more critters can fit through it. Properly installed 2x4 welded wire will stop birds of prey, dogs, foxes, coyotes, bobcats, adult raccoons, possum, and skunks. But snakes, rats, mice, many members of the weasel family, and maybe really young of some other critters might be able to squeeze through.
Just because something is possible does not mean it is likely to happen. I consider it highly likely mice will go through 2x4 wire to get to your coop. Snakes and rats are less likely but still highly possible. Snakes are mostly after the mice but can eat eggs or baby chicks. Rats are mostly attracted to the feed but can cause other problems. It is extremely hard to build a coop/run that can keep weasels out, especially the smaller ones, yet most of us don't have weasel problems. They just don't show up that often, but when they do it is really bad.
My run is basically 2x4 welded wire. Does it absolutely stop everything? No. But I've been satisfied with how it does work. I don't know if you will be or not.
