Here's my lay person thinking on this. By lifting objects up or down, you're altering the gravity wells of each object. A spaceship leaving Earth would make Earth's well a bit smaller.
The question you pose makes me think the following: if it requires no energy to fall down a gravity well, but you can use that falling to generate energy (hydroelectric dam for example), then where does that energy come from?
So this makes me think that as the following:
1. Increasing gravity wells (object falling) creates energy debt to the well (the debt is larger), in exchange for harnessed energy by things like hydroelectric dams.
2. Decreasing gravity wells (object being lifted) pays off the energy debt from the well by you using energy in the lifting action.
So for me, there's no energy stored, it's just a debt in the form of the size of a gravity well.
> A spaceship leaving Earth would make Earth's well a bit smaller.
While this is technically true, the effect is much too small to measure, and is not what is responsible for the usual phenomena we associate with gravitational potential energy.
The question you pose makes me think the following: if it requires no energy to fall down a gravity well, but you can use that falling to generate energy (hydroelectric dam for example), then where does that energy come from?
So this makes me think that as the following:
1. Increasing gravity wells (object falling) creates energy debt to the well (the debt is larger), in exchange for harnessed energy by things like hydroelectric dams.
2. Decreasing gravity wells (object being lifted) pays off the energy debt from the well by you using energy in the lifting action.
So for me, there's no energy stored, it's just a debt in the form of the size of a gravity well.