Not that deep, if you take the top layer of turf off about 6-10"" then level and apply a weed killer before covering in a weed suppressing membrane and cover with shingle to a depth of at least 4". Choose the larger 1/2" shingle the cats cannot use it like they can the smaller size. There are gravel calculators on line or the suppliers will calculate for you.
Dont forget you also need a frame/edging to contain the shingle if its a flat area 2x2 timber may be enough? If its on a slope you may need to look at wooden sleepers?