If there is no spacetime where would the earth "be" in the first place?

@user13267 that doesn't make sense. A perfect vacuum does not exist. You can't have a play without a stage, you can't have physics without the environment (which includes the space time).

If I'm not mistaken, wouldn't it be the mass of the object rather than the size of it that bends spacetime?

This explanation of gravity purely as curvature always leaves me deeply unsettled for two glaringly obvious reasons... it seems incompatible with conservation of energy and doesn't explain why two stationary bodies would move towards each other.

@Kaithar What do you mean? It is compatible with conservation of energy (as long as you are careful with your definition of energy), and it absolutely explains the attraction between stationary bodies.

@P.G.A mass does not cause the bending of spacetime. It is stress energy that bends spacetime.

@Kaithar With regard to the two stationary objects - imagine two cars running in parallel to the North which represents time (East-West represents space). At some point around north pole they will collide even though they did not 'move' - they path theough space-time led them to the same place.

@J.Murray being careful with your definition of energy is usually a sign of a flaw in a theory. But my point is that for an object to be accelerated by gravity a force must be applied, otherwise the kinetic energy just magically appears. Curvature alone isn't sufficient.

@Kaithar Being careful isn't a flaw. If you examine, say, the Schwarzschild solution to the Einstein equations, then for a particle of mass $m$ there exists a timelike Killing vector field corresponding to a conserved quantity wrt time translations which is equal to $\left(1-\frac{2GM}{c^2 r}\right)\gamma mc^2$. As your mass falls ($r$ decreases), the first factor decreases while the second increases, with the whole lot being conserved.

@MaciejPiechotka I can parse the analogy you're giving, but it doesn't work. Motion through time wouldn't impart motion through space... or more exactly, for two objects to move apart or together strictly because of space time without a force acting on them would imply that the fabric between them was expanding or contracting.

@Kaithar I may be misunderstanding you, but a Killing vector field is not a physical thing like an electric field or a gravitational field - it's just an isometry of the metric. Energy conservation is not always a meaningful concept in GR, but in static, weak-field cases (as we are familiar with in non-relativistic contexts), conserved quantities (e.g. energies) exist, and they reduce to what we'd expect in the appropriate limit.

@J.Murray so it's a pure mathematical construct? I think it's probably exceeding my knowledge in that area, sadly I mostly stop at applied math these days.

@Kaithar - but under GR the objects are NOT accelerating towards each other under a force. They are simply - in a somewhat newtonian sense - following a following a constant velocity locally straight line path - which because space time is curved is not actually straight so its called a geodesic instead. The space time curvature means the geodesics intersect, and transformed back into our normal co-ordinate system, and applied over time, the objects appear to accelerate towards each other as though influenced by an attractive force.

@Kaithar Yes. Recall from Noether's theorem that continuous symmetries of a problem correspond to quantities which are conserved along solutions to the equations of motion. In an analogous sense, Killing vector fields correspond to quantities which are conserved along geodesics. If there exists a time-like Killing vector, then we call the conserved quantity "energy." You can see a pretty straightforward explanation here.

@Kaithar follow ups here and here.