Yes, how big is "big" for you. And where are you located? The concern here is roof load, are you going to get a lot of ice or snow where the weight can break it down? If leaves pile up on it from trees it can get pretty heavy, especially if it rains and they get wet. This can happen with a wire mesh roof, snow, ice, and leaves can build up. If you put a solid roof on it it needs to slope enough that rainwater slides off. You need to be able to stand up in there to work. You probably knew this but I'll repeat it anyway.
To me one big issue is how wide an area are you trying to span. The wider that span the heavier the bracing across it has to be, either to support its own weight, snow, ice, or leaves, or even wind. If you just span it with wire the longer the span the more it sags, even without snow load.
Not totally sure what you mean by secure. Chain link, say dog kennels, are used a lot but snakes, mice, rats, and weasels can get through those openings. And watch gates and corners, you can get some pretty big openings there if you are not careful.
As far as how to build it there are all kinds of ways. I think Rosemary has some good ideas. You can build it on posts or using a foundation that won't rot. like concrete or cinder bocks and build it like you would a shed, just use wire for the walls instead of panels.
I don't have any good photos that show what I did to build my 12' x 32' run. I sank some treated posts into the ground and put cross braces at the top. I needed more slope. My roof, both metal and the 2x6 cross braces came from when a straight line wind blew the roof off of this shed so I had a lot of spare materials. The top opening between the coop (which is the end of shed) and the covered run has wire over it. I used 2" x 4" welded wire fencing to form the walls of the run. It has smaller mesh chicken wire on the bottom 18", mainly to keep baby chicks from getting outside. I used leftover hardwire cloth or 2x4 welded wire to make anti-dig aprons. I bought the posts and the wire. Most of the rest was cobbled together from scrap and leftovers.
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