Cut the cactus plant into small pieces or pads using a chainsaw. Cut horizontally, so the pieces slide off the cactus plant and land beside the plant.
Haul away the cactus pieces or cover them with a foot or more of fresh horse manure. The fertilizer will cause the cactus pads to rot away within six months.
Return to the roots and remove them using a spade. Dig deep down before applying weight to the spade to lift up the roots, as the roots can be growing deep in the soil. If the roots break in the process, ensure that you go back with a hand shovel to remove the remaining roots, as cactus roots will continue to grow.
Stir up the soil where the cactus was located, using a hand rake. Scatter some native grass seed into the soil and water it.