Push the cactus over with a shovel. Hold the shovel out horizontally in front of you and push the cactus away from you. Don't use too much force. You don't want to send needles flying. If the jumping cactus is too stout to push over, wrap a garden hose around the cactus (watch out for spines) and pull it down. Again, don't use any more force than necessary.
Scoop up all of the cactus parts with the shovel and place them into a wheelbarrow. Those little spiny, detachable arms will root if left in place. If the trunks are too heavy to be lifted with the shovel, sever them into pieces with the edge of your shovel blade and a little pressure from your foot on the shovel.
Dig the jumping cactus' roots up and toss them in the wheelbarrow. Try to get as many of the roots as possible. Dig wide, not deep. Cacti tend to have shallow, widespread roots.
Bury the cactus material 1 foot deep, or discard it.