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The concept of empty space is actually quite tricky, since our intuition "Space is empty when there is no particle in it" differs from the formal "Empty space is the unexcited vacuum state of the theory" quite a lot. The space around the atom is definitely not in the vacuum state, it is filled with electron states. But if you go and look, chances are, you will find at least some "empty" space in the sense of "no particles during measurement". Yet you are not justified in saying that there is "mostly empty space" around the atom, since the electrons are not that sharply localized unless some interaction (like measurements) takes place that actually force them to. When not interacting, their states are "smeared out" over the atom in something sometimes called the electron cloud, where the cloud or orbital represents the probability of finding a particle in any given spot.

This weirdness is one of the reasons why quantum mechanics is so fundamentally different from classical mechanics - suddenly, a lot of the world becomes wholly different from what we are used to at our macroscopic level, and especially our intuitions about "empty space" and such fail us completely at microscopic levels.

Second, consider the situation of two equally charged classical particles: If you only invest enough energy/work, you can bring them arbitrarily close. The Pauli exclusion principle prohibits this for the atoms - you might be able to push them a little bit into each other, but at some point, when the states of the electrons become too similar, it just won't go any further. When you hit that point, you have degenerate matter, a state of matter which is extremely difficult to compress, and where the exclusion principle is the sole reason for its incompressibility. This is not due to Coulomb repulsion, it is that that we also need to invest the energy to catapult the electrons into higher energy levels since the number of electrons in a volume of space increases under compression, while the number of available energy levels does not. (If you read the article, you will find that the electrons at some point will indeed prefer to combine with the protons and form neutrons, which then exhibit the same kind of behaviour. Then, again, you have something almost incompressible, until the pressure is high enough to break the neutrons down into quarks (that is merely theoretical). No one knows what happens when you increase the pressure on these quarks indefinitely, but we probably cannot know that anyway, since a black hole will form sooner or later)

So, in conclusion, one might say that at the levels of our everyday experience, it would probably enough to know about the Coulomb repulsion of the electrons (if you don't look at metals too closely). But without quantum mechanics, you would still wonder why these electrons do not simply go closer to their nuclei, i.e. reduce their orbital radius/drop to a lower energy state, and thus reduce the effective radius of the atom. Therefore, Coulomb repulsion already falls short at this scale to explain why matter seems "solid" at all -- only the exclusion principle can explain why the electrons behave the way they do.

The concept of empty space is actually quite tricky, since our intuition "Space is empty when there is no particle in it" differs from the formal "Empty space is the unexcited vacuum state of the theory" quite a lot. The space around the atom is definitely not in the vacuum state, it is filled with electron states. But if you go and look, chances are, you will find at least some "empty" space in the sense of "no particles during measurement". Yet you are not justified in saying that there is "mostly empty space" around the atom, since the electrons are not that sharply localized unless some interaction (like measurements) takes place that actually forces them to. When not interacting, their states are "smeared out" over the atom in something sometimes called the electron cloud, where the cloud or orbital represents the probability of finding a particle in any given spot.

This weirdness is one of the reasons why quantum mechanics is so fundamentally different from classical mechanics – suddenly, a lot of the world becomes wholly different from what we are used to at our macroscopic level, and especially our intuitions about "empty space" and such fail us completely at microscopic levels.

Second, consider the situation of two equally charged classical particles: If you only invest enough energy/work, you can bring them arbitrarily close. The Pauli exclusion principle prohibits this for the atoms – you might be able to push them a little bit into each other, but at some point, when the states of the electrons become too similar, it just won't go any further. When you hit that point, you have degenerate matter, a state of matter which is extremely difficult to compress, and where the exclusion principle is the sole reason for its incompressibility. This is not due to Coulomb repulsion, it is that that we also need to invest the energy to catapult the electrons into higher energy levels since the number of electrons in a volume of space increases under compression, while the number of available energy levels does not. (If you read the article, you will find that the electrons at some point will indeed prefer to combine with the protons and form neutrons, which then exhibit the same kind of behaviour. Then, again, you have something almost incompressible, until the pressure is high enough to break the neutrons down into quarks (that is merely theoretical). No one knows what happens when you increase the pressure on these quarks indefinitely, but we probably cannot know that anyway, since a black hole will form sooner or later)

So, in conclusion, one might say that at the levels of our everyday experience, it would probably enough to know about the Coulomb repulsion of the electrons (if you don't look at metals too closely). But without quantum mechanics, you would still wonder why these electrons do not simply go closer to their nuclei, i.e. reduce their orbital radius/drop to a lower energy state, and thus reduce the effective radius of the atom. Therefore, Coulomb repulsion already falls short at this scale to explain why matter seems "solid" at all – only the exclusion principle can explain why the electrons behave the way they do.

Third, the kind of force you need to create such degenerate matter is extraordinarily high. Even metallic hydrogen, the probably simplest kind of such matter, has not been reliably produced in experiments. However, as Mark A has pointed out in the comments (and as is very briefly mentioned in the Wiki article, too), a very good model for the free electrons in a metal is that of a degenerate gas, so one could take metal as a room-temperature example of the importance of the Pauli principle.

So, in conclusion, one might say that at the levels of our everyday experience, it would probably enough to know about the Coulomb repulsion of the electrons (if you don't look at metals too closely). But without quantum mechanics, you would still wonder why these electrons do not simply go closer to their nuclei, i.e. reduce their orbital radius/drop to a lower energy state, and thus reduce the effective radius of the atom. Therefore, Coulomb repulsion already falls short at this scale to explain why matter seems "solid" at all - only the exlusion principle can explain why the electrons behave the way they do.

Third, the kind of force you need to create such degenerate matter is extraordinarily high. Even metallic hydrogen, the probably simplest kind of such matter, has not been reliably produced in experiments. However, as Mark A has pointed out in the comments (and as is very briefly mentioned in the Wikipedia article, too), a very good model for the free electrons in a metal is that of a degenerate gas, so one could take metal as a room-temperature example of the importance of the Pauli principle.

So, in conclusion, one might say that at the levels of our everyday experience, it would probably enough to know about the Coulomb repulsion of the electrons (if you don't look at metals too closely). But without quantum mechanics, you would still wonder why these electrons do not simply go closer to their nuclei, i.e. reduce their orbital radius/drop to a lower energy state, and thus reduce the effective radius of the atom. Therefore, Coulomb repulsion already falls short at this scale to explain why matter seems "solid" at all -- only the exclusion principle can explain why the electrons behave the way they do.

Matter does not pass through other matter mainly due to the Pauli exclusion principle and due to the electromagnetic repulsion of the electrons. The closer you bring two atoms, i.e. the more the areas of non-zero expectation for their electrons overlap, the stronger will the repulsion due to the Pauli principle be, since it can never happen that two electrons possess exactly the same probability to be found in an extent of space.

Matter does not pass through other matter mainly due to the Pauli exclusion principle and due to the electromagnetic repulsion of the electrons. The closer you bring two atoms, i.e. the more the areas of non-zero expectation for their electrons overlap, the stronger will the repulsion due to the Pauli principle be, since it can never happen that two electrons possess exactly the same spin and the same probability to be found in an extent of space.