Space itself is 3-dimensional (or 4-d if you include time…and even higher dimensional orders if you want to get into deeper theories).

The reason almost all solar systems (including our own) are relatively flat, or disk-shaped, is because of the way solar systems are formed. Solar systems start as large clouds of gas and dust which coalesce into orbiting objects. The gas, dust, and objects “flatten out” as they spin around the central star because of the physics (conservation of angular momentum).

So even though the planets, asteroids, etc, in our solar system are all mostly moving along a relatively flat plane, the surrounding space is not 2-dimensional. Our planets also vary in how flat their orbits are. For example, Mercury’s orbit has a tilt of about 7 degrees from flat.