Depends on what you are trying to keep out and how you build it. If you don't put a top on it raccoon, bobcat, and foxes can climb in. Many critters can either dig under or just push their way under the fence, depending on how it is built. If your wire has really small holes it can keep out some really small critters but is typically so fine bigger critters can break it. Heavier wire that is harder to break usually has holes large enough that many critters can get through or reach through. Some fencing is much more expensive than others so how big it is might be an issue. There are all kinds of trade-offs.
Do not fall for the myth that critters only hunt at night. I've seen bobcats, foxes, coyotes, raccoons, a mink, even possum and skunks out hunting during the day. That possum really surprised me, I did not expect that. Of course, there are flying predators. For many of us dogs are the biggest problem, they can certainly be out during the day. But your risk is typically greatest at night when there is no human activity to scare these critters away. I use a philosophy of a predator-resistant run during the day and lock them in a predator-proof coop at night. It's worked for me so far but you never know.
I built my run out of 2x4 welded wire, heavy enough to keep out coyotes, dogs, fox, bobcats, raccoons, pretty much any big animal. I used smaller chicken wire on the inside on the bottom couple of feet. That was not so much to keep smaller critters out, snakes, mice, rats, and some smaller weasels can climb well enough to get above that bottom couple of feet. I did that partly to keep the chickens from poking their heads through the wire to eat grass outside where a predator could take it off, but mainly to keep baby chicks from getting outside and away from Mama's protection.
Another problem is that you don't know how good what you are doing really is until it is tested. You might go years before what you use is tested by a certain predator. I can think of one guy who used to be on the forum that used chicken wire. It worked for a couple of decades until two big dogs showed up and ripped it up. It could easily have gone another 20 years without a problem, but it didn't.
I don't know what the right answer is for you. My choice was to go with something on the larger side to stop the big ones but I'm still vulnerable to some smaller ones.
