The Core is a pocket dimension created by Gaia located in the center of the Earth. Only kiji can enter, although Gaia has the ability to lock them out. Currently only Gaia and Kokoro permanently reside there. Gaia does not have the ability currently to leave the core, although she can see what goes on in the world through the senses of the kiji and nature itself.
The Core is a massive, singular size, unchanging in dimensions. It is approximately 760 miles (or 1220 kilometers) in diameter, meaning it takes up all the space of the Earth's real inner core, but on a different plane. It is impossible for anyone to dig straight to the Core, due to Earth's outer core functioning as a super toasty (read: lethal) natural barrier. Kiji, however, are able to freely enter and exit the Core, thanks to multiple portal routes littered all over the world.
Despite the Core technically being the center of the planet, it is not inhospitably hot thanks to Gaia's power. If a human for example entered, they would not be incinerated, although painful sunburn is likely! The Core is centered in the middle of the planet because it is the most viable location to be able to spread energy throughout the world to help keep it stable.
The Core also has a unique way of dealing with gravity. There is a false, miniature sun-like orb in the center of this hollow dimensional sphere, and the walls of this sphere act as the floors, gravity behaving in an inverted manner.
There is vegetation, although somewhat sparse. None of it is made from the planet; all of it is created from pure energy, made to reside within the Core. That means despite it looking like there's plantlife, this is a semi-solid illusion created by raw energy. Should any living creature try to make a home here, they would die of starvation.
There is naturally no wind, unless generated by the kiji themselves or by Gaia’s presence.
Minor Points of Interest
Kokoro's den area, up near the central sun core. Although they don’t live inside the false sun, they’re so close that to most kiji, they might as well. Kokoro’s den is largely rocky, and there’s some false plant life to help break up the monotony. There is a small, private alcove where they like to rest. Where Kokoro often likes to lounge is a raised boulder, a bit higher than the rest of their abode, and many kiji refer to this as Kokoro’s ‘throne,’ though it is not shaped like one.
Three small 'village' type locales, inhabited by visiting kiji who are coming to rest after depositing energy. They recuperate and socialize here. These ‘villages’ are cleared areas made up of pits, alcoves, small caves, tents, or den-like structures. One village has something resembling an arena, where spectators can watch other kiji “resolve their differences.” It’s almost never a serious fight that occurs in this arena; oftentimes it’s kiji comparing their skills or features and bragging about themselves.
There's two pools of water that kiji often frequent for summoning purposes, making them hotspots. Larger numbers of kiji tend to gather at these pools for finding other summoners or if they need advice from other kiji on how to summon. Those who like to summon frequently also find themselves at either of these pools.
There's a cave where kiji seclude themselves often, for rebirthing in relative privacy. As rebirthing isn’t being done constantly by everyone, while there’s a bit of a queue, generally privacy is guaranteed.
As for Gaia's preferred abode, she doesn't technically have one. Due to how she built the Core, and the events leading up to and during the Thinning, she had such little energy that she ended up becoming the Core itself. She does not have a physical, visual appearance, save for the environment itself. Under rare circumstances, she can take on a brief humanoid manifestation when called upon by Kokoro or other valued kiji.