Mirage Tower
The Mirage Tower is a tower in Pokémon Emerald made of sand. Note that it's not to be confused with Mirage Island. Mirage Tower will appear after speaking with the Ruin Maniac just outside of Fallabor Town in the house on Route 114. Then, go to the Desert on Route 111 and you should see it. If you don't see it, it may take some time to appear. Do something else and check back often until it does.
Inside the Mirage Tower
Be sure to have the Mach Bike and a Pokémon that knows Rock Smash to get around inside. Proceed to the top floor where you'll find the Root and Claw fossils. Choose one, and exit. The Mirage Tower will then crumble into the sand. You can still get the other fossil you didn't take by going to the Dessert Underpass later in the game.