How can we be so sure that atoms are 99.9 empty spaces?

You can bet it's not. Bohr's model was very popular when physicists first began studying atomic structure. It sees electrons as planets and atomic nuclei as suns. Great for hydrogen, not much else. Even hydrogen has problems. First, physicists have to assume that electrons move within a set distance, unlike planets and everything else. Second, if you force electrons to take a curved path, they emit radiation. The electrons lose energy and spiral into the nucleus. For some reason, physicists have to assume they don't.

Forget what planetary models you've heard about. This is not right. It's a shame that it works so well on hydrogen because we still have terms like orbit and spin. At least when physicists talk about the taste, color, and charm of quarks, people don't think they actually taste like chocolate, or blue, or frivolous.

There are many illustrations of the track itself. This is an attempt to show what they look like in atoms. This is argon. There is an inner orbit with only two electrons, another spherical orbit with two electrons, then three lobes with two electrons each, and then an outer shell of a repeating structure. So where are the electrons? Anywhere on track, any time.

How could they all be in the same place? Imagine a concert hall. Where are the notes on the violin? It's all over the hall, all the time. Where are the trumpet notes? It's all over the hall, all the time. Where are the notes on the flute? It's all over the hall, all the time. This all happens at the same time.

The planetary model of electrons surrounding atoms is very similar to imagining small visible musical notes floating from an orchestra through the air to your ears.

So there is no empty space in an atom. There's also something called a Pauli exclusion, which prevents particles from occupying the same space if they have the same quantum properties. Trying to move one atom through another, Pauli repulsion will not allow two identical orbitals to overlap. Why? That's it for the time being. Time is described as God's way of preventing everything from happening at the same time. Pauli incompatibility can be thought of as God's way of preventing everything from happening in the same place.

Some people think that electrons are waves, while others think that electrons are particles. All, none. They don't have a good analogy with everything on the atomic scale, and we don't have a good analogy with the scale of our own existence. When you fire electrons at an old CRT TV screen, it's very efficient to think of the electrons as particles. When you use an electron microscope, the wave analogy is better. When you study orbits or design microelectronics using quantum mechanics, your only option is to delve into mathematics.