Is it possible to run a Unity 2D game within a Unity 3D world?

If so, how would I accomplish this? I essentially want to run it in the game world on a plane. Any tutorials/videos you could link would be greatly appreciated.

Yes, unity is a 3D engine even when you make a 2D game, so all 2D games made in unity are in reality 2.5D, the closest thing to making a pure 2D game would be doing it all in the canvas, you might be able t do this with canvas in world space mode, but you can also do it all with sprites, just parent them with the plane program the correct behaviours.