Is the Earth in space, or is space in Earth?
Space is part of the coordinate system that we define position and distance by. Time being the other part of that coordinate system.
The Earth exists within this coordinate system, and it also occupies a volume, which means that there are plenty of definable locations within that volume. If there was "no space" within the Earth that would mean that it has no volume.
However, sometimes when people colloquially refer to "space" they are specifically talking about "empty space," i.e. vacuum or near-vacuum. If this is the kind of space we're talking about, then obviously the Earth exists in a near-vacuum but there is no near-vacuum within the volume of the Earth, as the Earth is quite dense. I say near-vacuum because even what we call "empty" space isn't really all that empty, since the space surrounding the Earth is filled with radiation and charged particles from the sun. Even interstellar and intergalactic space has atoms and molecules floating around.