The Shinigami Realm is where Shinigami live and dwell throughout their existence. The Shinigami Realm is connected to the Human World by a Portal through which Shinigami alone are able to travel. They are able to remain in the Human World as long as their Death Note is in a human's possession.
The world that Shinigami reside in exists above the Human World. Ryuk has to travel underground in order to reach the entrance to the realm of humans underneath. This is further proven when Ryuk is first seen in the Human World after leaving the Shinigami Realm, flying down from a higher altitude than skyscrapers.
Appearance and lifeforms
The realm is dark and appears lifeless. Very little vegetation is found in this barren land. Plants are never shown, though they do exist, as Ryuk brings an apple from the Shinigami Realm to the Human World, although it was shriveled and unlike apples that grow on Earth. Sentient beings other than the Shinigami do not appear to exist there.
The realm includes different landscapes and does not have modern infrastructure. Shinigami tend to congregate in areas with bone and/or stone structures. There are also many chains, surrounding both common areas and the Shinigami King's cave, where there are hundreds of chains with hooks on the end. There is at least one mountain with a cave structure where Ryuk is living, which is shown in the one-shot special of the anime. This mountain could also possibly be where the Shinigami King resides.
The realm is surrounded by deserts that have various strange objects. The spheres, which are shaped like concentric orbs, are littered about the surface. These artifacts have openings which let Shinigami see anywhere in the Human World from a birds-eye view and are what allow Shinigami to take human lives with their Death Notes without traveling to Earth. When or how the objects were created is currently unknown.
Ryuk explains that there is not much to do in the Shinigami Realm. Shinigami usually gamble with one another, watch the events occurring in the Human World through the spheres, and terminate human lives with their Death Notes in order to prolong their own, though this is considered work and Shinigami are ridiculed by others for it. They also pass time by eating, though they have no need to. The lack of stimulating activity is Ryuk's primary reason for dropping Sidoh's Death Note into the Human World.