Dogs will dig into a run, domestic/feral cats are not so much a problem for grown standard hens. Raccoons are the smartest and most persistent predators I have to deal with. There were 3 BIG ones eating the cat food I accidentally left out last night after dark! Coons will reach through or chew through chicken wire; they'll climb, open latches, and otherwise amaze with their abilities. Determined dogs will chew through chicken wire too.
My 260 sq ft run may be somewhat over-engineered, but I do have 2 feet of 1/2# hardware cloth around the entire bottom of my run. I have an "apron" of 2" x 4" welded wire fencing going out at least 18 inches, and a bit more in most places--mostly because I bought 4' and 6' tall fencing and cut it down the middle with snips to use as the apron. The 6' tall run above the hardware cloth is 2" x 4" welded wire fencing secured to 4 x 4 posts and 2 x 4 rafters. The top of the run is also covered with welded wire fencing because the coop/run is under a big Doug fir tree, and I don't want coons "dropping in". I joined the wire with heavy duty zip-ties. Chicken wire and bird netting won't do anything to deter a determined raccoon.
The coop is raised up about 3 feet and part of the run is under the coop. The exterior nest box is secured with 2 different closures, and the "people" door is actually a whole side of the raised coop that is hinged up that I keep a carabiner on, though I think it's too heavy for any critter to open without anything to stand on and get some leverage. All ventilation is covered with 1/2" x 1/2" hardware cloth.
I spent nothing on the play structure I used for the coop, and used wood and materials that I either had or found for everything EXCEPT the wire/fencing and connectors (screws and zipties.) Even the 4 x 4's for the run posts were cut up from the cedar structure of the swingset that was attached to the raised playhouse that I used.
I actually have been leaving the door from the coop to the run OPEN in the summer, as I am pretty sure about the security of my run, but will have to close it when it gets colder around here. The ladies will not be happy.
