While not a perfect definition of "real", don't we already have images of atoms?
If we adopt that same definition for spacetime, we also have photographs of it, e.g. https://en.wikipedia.org/wiki/Solar_eclipse_of_May_29,_1919#... and https://en.wikipedia.org/wiki/Gravitational_lens#Gallery.
What does this even mean? Unless you're talking about some supernatural "atomic sense", there's obviously no way to "sense them directly" - we sense them through their interaction with other things combined with our model of those things and the interaction.
I think atomic phenomena may feel somehow more familiar because it takes far less energy to make them dynamic than it does for gravitational phenomena. If we were somehow huge beings made of binary stars or something, gravity would probably be the very familiar force and atomic phenomena almost inconceivably small to you.
Basically you start with some base information(axioms and postulates), then you gather some experimental evidence for those axioms and postulates. You establish some rules on which your experimental and observational universe works. Then what follows is volumes and volumes of Math.
If Im not wrong relativity itself was developed over a decade. And I'd assume it took tons and tons iterations, retries and tweaking to make it work.
Where common people like us get lost is looking at Symbol manipulation directly. At that stage Math scares the day light out of us.
The right approach is to start with your own axioms, postulates, truths and symbols and start working from there.
This is really like some newbie seeing a super massive code base and thinking it could be outside their reach to ever work with such a thing. But if you look at it carefully. All there is to it is loops, condition statements, operations, variables and basic abstractions.
> You sense spacetime directly too. It's what holds you do this planet we inhabit.
Nope.
At the same time not every real image is real. We could sometimes see two Suns in the sky, or we can see a rainbow, but it doesn't mean that the second Sun and the rainbow are real.