Where does the Earth end?
The surface of the Earth is the continuous surface of a 3D sphere and has no "end".
The space part of spacetime is the surface(usually called a volume) of a 4D volume (sphere? I dunno.) It might be a bumpy lumpy volume like the Earth's surface has topography, but it is thought to be continuous. It's hard to picture a 4D object as our brains are tuned for 3D existance so I personally rely heavily on the surface of the Earth analogy.