The Lunar Fortress is a crystal palace located on the Moon.
The fortress has since become her lair, though she can only visit it when she has the moon crystal or finds other ways to reach the moon, like hitching a ride in PJ Masks Headquarters, or either being towed in a moon bubble or getting teleported by Motsuki.
The fortress has, among other things, a throne for Luna Girl and various statues of her. In "Moon Madness", she upgraded the fortress with a mega magnet, capable of generating a moonbeam that could pull objects from Earth to the moon. At the end of the episode, though, the mega magnet was destroyed by the PJ Masks. Then in "Heroes of the Sky", the fortress was temporarily destroyed when Romeo ordered the Fly Bots to steal the moon crystals that kept the fortress stable.
It seems that the inside of the Lunar Fortress has oxygen inside of it as Luna Girl doesn't have the pink aura around her when she's inside. However, the PJ Masks still wear their spacesuits when inside.