First insight that is useful here is the fact that you can make a side of the fence any multiple of 2 feet, as long as it's at least 4. By taking 4 foot pieces, you can make 4,8,12, and so on. If you start with 6, then take 4 foot pieces, you can make 6,10,14 and so on. You don't need the 8 foot pieces, they're only a convenience.
So, suppose one side is the smallest possible, ie., 4 foot, what must the other side be to satisfy the constraint?
It must be greater than 216/4=54, so at least 56 (remember, only even sizes), and smaller than 480/4=120, so at most 118.
I've plotted all the possible widths and lengths, but there are a lot of them! (See picture, green area).
Selecting which fence sizes you need is possible using the fact that you can create any multiple of 2 starting at 4. I hope this gives you some pointers!
ways.