Place a tarp on the ground next to the saguaro cactus. The tarp will help you haul the cactus away when you are done removing it from the ground.
Insert a sharp-edged shovel into the soil around the cactus, creating a 1-foot diameter circle. Insert the shovel as deep as it will go so you cut the main roots of the plant.
Wrap a garden hose around the cactus and have a helper stand behind the tarp, holding the hose. Insert the shovel under the cactus on the opposite side of the cactus from the tarp.
Pry up on the shovel as the helper pulls on the hose. Keep prying until the cactus falls onto the tarp. Wrap it in the tarp and carry it to the trash. If it is too heavy to haul or carry, use a hedge trimmer or chainsaw to cut it into manageable pieces and remove them one at a time.