You don't need a very high fence to keep rabbits out, waist high would
be adequate. Use small mesh and bury the bottom at least eighteen inches
in the ground.

Our cat and two jack russels would love to see more rabbits in the
garden, so could you get a harder cat?

Depending on the location of other gardens and houses, an air rifle
would do the trick nicely as well, although I suspect this is not an
option in your case.