Making 100% gopher proof beds that will last for twenty years isn’t difficult, but can involve quite a lot of materials and get quite expensive. Though when spread out over 20 years this isn’t too bad and they will quickly pay for themselves in peace of mind. If you are a good scrounger you can probably find ways to make them for less.
You can use anything for the sides of the beds, but where I live the material of choice is rough sawn redwood 2 x 12, as it will last for 20 years. I don’t really approve of this, as it is a waste of redwood trees and have never bought new wood for this purpose. I usually use old 2 x 6 boards salvaged from old decks, one on top of the other to get enough height (this helps to prevent the gophers from simply walking on to the beds and digging down into it).
The gopher protection is provided by 1/2” galvanized hardware cloth (plastic coated galvanized would be even better, but is hard to find). This should not just go on the bottom of the bed however, for maximum protection it should go up the inside of the bed, to within an inch of the top. Essentially you are creating a large wire basket that completely fills the frame. This means that the bed will be 20” narrower than the width of the wire. If you want to create wider beds you will have to use two pieces of wire to get sufficient width.
When cutting the wire pay particular attention to the corners, you don’t want any holes. If I’m in any doubts as to whether I have made a good corner I reinforce it with a second piece of hardware cloth. I staple the wire to the frame with a pneumatic stapler, to ensure gophers can’t force their way up between frame and wire.
The beauty of this system is that it doesn’t rely on the frame for protection, it is simply there support the wire. If you use inferior wood and it starts to rot away, your plants will still be perfectly secure (you could simply put another piece of wood on the outside, without disturbing the bed). This method also solves the problem of gophers simply chewing a hole in the side of the bed. If they do they meet cold hard wire, rather than delicious succulent vegetables.