Wear thick gloves and old, thick clothing that covers the body. Then, cut down the yucca with a pair of loppers and a hand saw, if necessary. Dispose of the plant parts in the trash in a thick, construction bag doubled inside another construction bag. Do not compost a yucca.
Dig out the roots with a spade or shovel. Dig as deep and wide as you can to get every last root. Then, go through the soil with a sieve, if possible, to find and dispose of all the rhizomes or pups.
Fill the hole with soil and if the yucca grows back, the roots are too deep for you to dig out. Cut the new growth back to the ground and immediately paint on a non-selective herbicide such as glyphosate with a paintbrush. You can also cover the area with six layers of cardboard or newspaper and then cover that with mulch to hide the layers. Replace the cardboard or newspapers as they decompose for at least eight months to starve the yucca roots to death.