There are two key points to keep in mind when trying to rabbit-proof your garden or lawn. The first is that rabbits love to burrow and the second is that they are able to squeeze through openings that appear smaller than their bodies.
Part of building a fence always involves deciding on the height you want the slats or chain link and posts to be. That in turn is dictated by your preferred look, local codes, the cost of materials and so forth. Nevertheless, be sure to add an extra foot not only for the posts but for the slats as well.
That will make building the fence a bigger effort. Sinking posts is difficult enough, but there is normally only one for every eight feet or more. Sinking fence panels or chain link a foot or so into the ground is certainly more difficult than simply laying fence across it. Whether the effort is worthwhile depends on how serious you are about keeping the rabbits from digging underneath.
The second aspect of building a rabbit proof fence is equally important. Rabbits can squeeze through an opening only a few inches wide, small ones even less. Keeping them out requires spacing the slats close enough to prevent access.
Here again, the cost of materials, the effort required and your intended design will influence how close together your fence slats will be. Chain link is a standard size, and will keep them out, so it is not an issue here.
One way to offset both a higher cost and maintain a desired look is to space slats however you like, but supplement the lower two to three feet with mesh or chain link. Rabbits do hop far, but they can only leap up a foot or so to get through a fence.
A mesh can also add a nice design element at only moderate cost. However, bear in mind that rabbits can chew easily through any plastic or cloth. If a rabbit proof fence is to be of any value in this situation, it needs to be a sturdy material, usually metal. To keep the expense down, an alternative plan is simply to limit the extra protection only to gardens and flowerbeds. Planting an out of the way section of yard with clover will encourage them to favor that area. They will still feed elsewhere, but it improves the odds. That will reduce the number and height of fence slats or chain link required.