It is not possible to capture Mewtwo in "Pokémon Emerald." The easiest method of capturing him and getting him into the game is to capture him in "FireRed" or "LeafGreen" and transfer him over.
- Play through the story mode of Pokémon FireRed or LeafGreen
Unfortunately, if you haven't played either of these games before, obtaining Mewtwo will not be a quick process. To get access to the cave where Mewtwo hides, you'll have to play through either game until you've defeated the Pokémon League and are in the Hall of Fame. You'll then need to complete the Sevii Islands quests and restore the Network Machine along the way. Once all of this is done, the man guarding the entrance to the Cerulean Cave will allow you to pass.
- Explore Cerulean Cave and defeat Mewtwo
The entrance to the cave is just outside of Cerulean City. The cave is a maze guarded by a number of different Pokémon types that are very strong, and also contains a number of treasures. The Surf ability will be required to reach Mewtwo, who waits on a platform at the end of the cave surrounded by water. Mewtwo is level 70 and uses Psychic attacks, so you should be adequately prepared. If the Master Ball has not yet been used, it can be used to capture Mewtwo when he is weak.
- Transfer Mewtwo to "Pokémon Emerald"
After obtaining Mewtwo, he can be transferred directly to your copy of "Emerald" by way of the Network Machine.