I’m thinking at least 2’ above ground and maybe 6" deep?
Probably a coated 1" grid type of fencing.
Will this work?
I think the garden will be 15’ x 20’.
Ground is fairly soft, so hopefully shallow trenching won’t be too hard.
The advice I got back in 2020 when I inexplicably had lots of extra time at home was to make it 3-4’ tall, and also flat on the ground for 2’ around the perimeter. Rabbits (and, for me, groundhogs) are excellent diggers, and can dig right under a fence–but they won’t try to dig until they get to the fence. If you have flat fencing on the ground, covered by mulch, they’ll try to dig right there, hit the fence, and be stymied.
It’s like a moat, only instead of water, it’s wire.
Chicken wire. Embed the bottom in the ground or fold it down flat 6 inches outside the garden. I put up plastic coated grid wire as high as needed to stop other invaders and then a layer of chicken wire over that. Very difficult to keep rabbits and other diggers out. You can’t stop squirrels and you need pretty tall fencing to stop deer.
ETA: 2 feet out as suggested above is even better.
This is what I did last spring (the fence around the main vegetable garden is about 2 1/2 feet tall). No rabbit incursions yet.
I got bunnies in my kale and tender greens as we speak.
I hate those little bastards and their teeth marks on my leafy greens. 
I have bunny wire all around the garden spot. 3 feet. Then chicken wire is buried.
Above is electric fencing.
hidden by Mod
And a 22. with a good aim, helps.
No need to worry about deer, squirrels are pretty much impossible to keep out and I won’t try and groundhogs unlikely thankfully, I hate those little bastards.
Please keep the gun crap out of my thread, that is off topic to a static fence.
2’ high poultry wire for the rabbits attached to 48” high cattle panels for the deer/humans.
Never buried the poultry wire as I had no evidence of any tunneling critters at the time I put it up years ago. Last summer I got moles/voles in the yard, no evidence of damage in my garden yet. My have to redo the poultry wire with partially buried hardware cloth (smaller holes) if it becomes an issue.
So sorry. Did not mean to offend.
I bought myself a 100 foot (?) roll of chicken wire about 20 inches high. It’s nice & pliable and the vinyl coating still looks good about 10 years later. I used it along the bottom of the the chainlink like the others describe to help manage rabbit traffic patterns.
I’ve used the roll for other things, it’s sort of handy for holding the shape of bags and stuff. The zipties I used to secure it to the fence gave up, understandably, only after a year or two. What have lasted well are aluminum rods ‘woven’ through and secured with wire.
My rabbit wire is secured with hog rings. There’s a tool you use to open them up, you place it where you want and release and double wire loop connects to two fences. Nothing has ever chewed thru them. That I noticed.
If you think you might want to remove this use a less permanent connector.
Nylon ties.
How did you train 'em to hold the epee?
Ninja’d! Damn you 
That’s a dirty trick, the rabbits can’t eat with the tiny, custom made with ear holes, face cage.
Cruel 
I had one rabbit eating all my lettuce and other things. I asked my gardener and he simply put a “fence” around the garden using some 6" wide green netting. It seemed obvious that any self-respecting rabbit could have hopped over (or even dug under) but this one didn’t. And the gardener didn’t expect it to. YMMV
Should have mentioned that I hammered 10-inch metal landscape pins into the ground to hold down edges of the six-inch stretch of fencing that lies flat at the perimeter. It further discourages digging by the rabbit bastids.
post hidden by Puzzlegal
But they taste so good! A rabbit and Guinness pie is delicious. Butchering a rabbit is a bit of a pain, especially because I had never done it before. Youtube to the rescue. The trick is to poke the ribs through the meat to remove them.
A hearty glass of Guinness and some veg, then around an hour stewing, before placing in a deep pie dish and covering with pastry. Bake until the pastry is nicely coloured.
You will soon come to welcome your rabbit neighbours.
Is there any way to fence off racoons? And does anyone know how they taste?
Moderating:
Please keep this thread on the topic of rabbit fencing. Please do not discuss lethal methods of dealing with rabbits, nor any of the potential subsequences of those
I’ve found that rabbits are lazy, unless they are very hungry or you are fencing off something they really love, like peas. So a lot of my rabbit fencing is just 8" of anything that gets in their way. That’s usually enough for them to decide to eat my grass, instead of whatever is inside the fence.
But i agree that if they want to get under a fence, they dug very well. And extending the fence horizontally, just under the ground, is a good way to deter that.
I use hardware cloth, but my more troublesome pest is chipmunks, which walk right through chicken wire. If i only had rabbits, I’d probably use chicken wire, which is cheaper and easier to deal with than hardware cloth.
I have a few spikes holding the chicken wire down, along with rocks, bricks, mulch, and assorted dirt, sticks, and other other yard detritus over the chicken wire on the ground.
Mentioned above by @Left_Hand_of_Dorkness is the idea that a layer of mulch on top of the wire fencing will cause rabbits to give up trying to dig through it when they hit the fence. I can’t verify that from my own experience but it makes sense.