Below the Planck distance, "closer" ceases to be a meaningful concept. Similarly, distances larger than the size of the observable universe may also be meaningless.
We don’t know if that is the case. That is only one possibility.
But it seems very clear that whatever happens at the Planck distance or lower isn’t simple smooth space as we model it for larger scales.
The “relativity” aspect will almost certainly still apply in some way, and still form an emergent basis for the special relativity effects you point out.
All of general relativity has to be emergent from the to-be-discovered laws of the underlying small scale structure of space.
(I.e. general relativity isn’t wrong, it is just not complete. Similar relationship as with Newton’s Law of Gravity, which was also correct, but breaks down beyond the conditions it covers well, because it was not complete.)
The smallest scale is also where we expect more “light” shed on the initial conditions of the universe and potentially the insides of black holes. Two other conditions where general relativity already breaks down.
It doesn’t that any particular phenomena we understand today will break down. Just we will be able to see richer behavior, that has been there all along, than our models cover today. And so be able to understand more conditions than we do today. And perhaps new capabilities to engineer things than we have today.
Obviously quantum field theory, as it exists today, breaks down at the Planck distance too, since it can’t tell us what is happening at smaller distances either.
This isn’t the least bit controversial or surprising. It just means that even with both theories, we can’t explain everything yet.
When we do have an accurate theory of the fine structure of space and time, it will also be a theory of the fine structure of fields. Since space-time is integral to both theories.
It may even be the long sought after unification.
But for distances much larger than the Plank distance, the fine structure theory will still simplify into the GR and QFT we have today.
Just as GR under many conditions we encounter every day, further simplifies to Newton’s Law of Gravity.
For examples of the "good" kind, Newton's laws of motion are indeed just an approximation of special relativity. The Schrodinger equation is just an approximation of QFT.
In contrast, GR came in and showed that Newton's law of universal attraction is completely wrong, at the fundamental level. Sure, it predicts certain phenomena correctly, but so did the epicycles that it replaced. Similarly, QM/QFT showed that Newton's laws of motion are also completely wrong, that objects (or at least particles) don't even move according to some laws of motion, they are only described by a probability wave that moves according to some laws, and between two interactions they have no definite state and are not even localized.
And of course, GR and QFT disagree on these parts - GR generally agrees with Newton's laws of motion (with the SR corrections), and QFT generally agrees with Newtoninan gravity. But you can't use GR's gravity with QFT's laws of motion, so we know one or both are broken. A new theory is very likely to also "overrule" one or both of them and show that they are not just an approximation, but completely wrong, working only "by accident" on the scenarios where they have been tested. Especially if the new theory requires there to exist fixed space distances that all relativistic observers agree on.
People (some people) think that the universe really is that way, at the Planck distance. Actual experimental confirmation is somewhat lacking at this time...
I've also heard that there's nothing special about planck length other than it being universal constant that we and any conceivable aliens would agree on as a standard of measure. So, idk.
This is just according to our current theories, and you don't need any speculative ideas about granularity of space. Mathematically, there's no problem in considering arbitrarily small length scales, because current theories are based on continuous space and time dimensions, but since we can't give physical meaning to small length scales, that's a clue that a more fundamental theory will somehow not be based on continuous space and time
To my understanding there's nothing special about it except once you get to quantum field theory, and specifically spin-2 particles, which are the ones that hypothetically carry gravity (i.e. gravitons). (They're hypothetically gravitons because mathematically they result in conservation of the "stress-energy tensor", which is the same thing that Einstein's field equations give. And this isn't coincidence; it's because both are second-order field theories preserving Lorentz transforms aka special relativity). But notably if you only get through non-relativistic quantum mechanics, there's nothing special that happens at Planck length.
Planck length becomes special in QFT because at Planck length, the feedback (my force on you affects your force on me affects my force on you etc) of the quantum field mathematically explodes in something called "UV divergence". Notably, this also happens for other force carriers, not just gravitons, but a mathematical trick called Renormalization fixes that. Renormalization is based on shady techniques like assuming the sum of all whole numbers is -1/12. (IIUC this is a fairly straightforward pure mathematical result from complex analysis, required to ensure consistency of Fourier transforms for infinite sequences, known as analytic continuation / Riemann zeta function. Of course it makes no sense that nature should behave that way, but one day some physicists thought "let's try this weird math" and it matched experiments). However that technique only works for lower-spin fields (E&M, weak, strong) because the feedback is linear, hence sum(1+2+3+...). But in spin-2 fields the feedback is (quadratic? exponential?), and the same trick still gives an infinite result.
Ultimately one could say this is the only thing standing in the way of a complete theory of quantum gravity. If we could get the math for gravitons not to blow up at tiny scales, we'd be done (experimental verification notwithstanding). String theory is one attempt: "Let's say there are no point particles, only strings, and this problem goes away". Other approaches exist too. But ultimately right now, there's no way to say for sure whether sub-Planck exists, doesn't exist, exists in some ways but not in others, or whether all of physics is wrong and we have to start over.
A quick addendum: Planck energy is approximately the chemical energy in a tank of gas. Planck mass is about that of an amoeba (so an antimatter bomb made out of an amoeba would explode like a tank of gas (Hiroshima was 0.6 grams of mass), a photon with that energy would have a Planck-length wavelength, and an amoeba-mass black hole is Planck-length). Given there's nothing overly special about Planck energy or mass, my money is on there being nothing special about Planck length either.
Hopefully at least some of the above is correct.
https://www.youtube.com/watch?v=-OxVsVUesSc
About the Planck length, while the length itself might not be physically significant, black holes are definitely physically significant, and according to the current thinking there will be some smallest length that can be probed before what you're looking at becomes a black hole. I guess I was lazily assuming that would be around the Planck length. I would have to look into it more to come up with a calculation for that