Deer can jump pretty high or they can jump fairly wide but they don’t do both at the same time. So a little height plus some width works really well to keep them out of your garden. Personally I’d want the run portion tall enough that I can walk in there without bumping my head. I don’t know how wide you plan to make it but if you make it tall enough to walk in you should be pretty safe from deer eating your garden.
I don’t know how big your garden is but you are probably putting up a lot of fencing. That can get pretty expensive. You want to keep the normal predators away from your chickens but also raccoons, groundhogs, and rabbits out of the garden. Raccoons, possums, foxes, wildcats, groundhogs, and others can climb. You want to keep your chickens contained so they don’t destroy your garden. You have a few challenges.
To reduce costs I’d suggest you consider an electric fence. I use electric netting but I don’t think that would work for you. You’d need strands of electric wire on insulators at different levels on your outside fence, probably one pretty low to stop ground level and digging critters plus a couple higher up to stop climbing critters. One of those properly placed might keep your sheep off the fence. If you do decide to pursue this I suggest you call the ladies at Premier Fencing and chat with them about this. I’ve found them to be pretty helpful. Talk to them about how you manage gates too. Gates just might be your weak point.
If I were doing this I’d probably use 2” x 4” welded wire as the base for my outside fence. That’s strong enough and with a mesh size to stop most larger predators. I’d use the 4x4 posts at the corners, properly braced, but metal T-Posts as intermediate posts. You can do some searches online about fence building to see how to properly brace a corner post and spacing and such for the intermediates. You’ll probably need some pretty tall T-posts but they come in different lengths.
Another possibility if you get lucky is to find chain link fencing or maybe dog kennels on Craigslist to use. If you are not concerned that much with appearances you might be able to patch a fence together with those free materials.
I’d probably build the rest out of chicken wire or some type of finer mesh cheaper wire. The heavier the gauge the longer it will last, but your goal for this wire is to keep the chickens in and flying critters out. The outside electric fencing will keep the other critters away. I would cover it, not just for flying predators but because chickens can get over a pretty high fence if they want to. You don’t want your chickens loose in your garden. They can be worse than deer.
You may find it necessary to use an apron, either to keep rabbits or some other critters from going under your fence or to keep the chickens from finding a way under that inside fence into the garden. An apron is where you take a 12” to 18” wide piece of fencing like chicken wire or in your case some other finer meshed wire for rabbits and lay it flat on the ground, then attach it to the bottom of your fence so nothing can get through. Normally you don’t have to bury it, the grass will grow up through it and hold it down, but a typical installation is to remove the sod and then put that back on top, essentially burying it about 2”. Depending on the size of your moat and the chicken density they may or may not keep the green stuff out of there.
There are some downsides to an electric fence. You have to keep the grass and weeds from growing up into it and shorting it out. That can be a pain because a weed-eater can damage the electric fencing. You have to be careful and it takes maintenance time. Another potential problem is when the electricity goes off you lose protection. It also does not handle snow well if the snow covers a hot wire. It can short it out. Once a predator is bitten by that fence it normally leaves the fence alone so this is not a huge problem but there are new predators occasionally coming around that have not been bitten. That’s another reason for a strong outer fence.
You also may have this issue, depending on your chicken density, though this has nothing to do with it being electric. I have an area about 40’ x 90’ inside electric netting. My chickens do not keep it stripped bare and they only eat certain vegetation. The stuff they don’t want plus even the good stuff sometimes gets so tall I have to mow it. It can become a jungle in there, especially with stuff they don’t eat. That’s another reason to have the top high enough for you to walk in there without banging your head. You may be walking through that weed-eating the weeds and grass.
I don’t know if this will help you or not but maybe it will give you some ideas. Good luck!
The moat's width would most likely be 5.5 feet. 
