Space, so people keep telling me, is empty. And even when it comes to material that we think of as observable matter, for example the earth, most of the atoms that comprise it are themselves mostly filled with space.
What percentage of observable space is actually observable matter? Or put differently, what percentage of observable space is space?
There are a number of ways this question could be answered.
Continuing Pablo Diez Silva's answer, according to quantum field theory all space is field with quantum fields. The values of these fields represents what is at each point in space. If the fields are $0$, nothing is there. But the fields are always fluctuating. This is often described as space being filled with virtual particles that wink in and out of existence. This gives empty space an energy.
Aside from this, space has stuff in it that isn't ordinary matter. Space is filled with light. Not a whole lot. But no matter where you are in space, you can see stars because the light from those stars is there. There is also light at wavelengths you cannot see. Stars emit UV and infrared light, and other wavelengths. The Cosmic Microwave Background is very long wavelength light emitted by hot matter at the beginning of the universe before there were any stars.
There are far more neutrinos in space than photons. Most of these were generated at the beginning of the universe. Many are generated in the Sun. These are nearly undetectable. They fly through ordinary matter without interacting, like light through a window.
There is ordinary matter in space. But it clumps. We live in a clump. It is spread out very thinly on average. The farther you get from a clump, the thinner it gets. A nebula is a relatively dense spot in a galaxy. The densest nebula has 10000 atoms per cubic centimeter. This is a vacuum 10 times less dense that the best we can make on Earth. Between galaxies there might be 1 atom in a cubic meter.
We can see gravitational effects where we don't see matter to generated them. Something must be there. It is called dark matter. To account for the effects, there must be 5 times more dark matter than regular matter. Nobody knows what dark matter is made of.
Another gravitational effect alters the expansion of the universe. A second mysterious thing must be present to account for it. This is dark energy. Again it must be there, but nobody knows what it is. There must be 4 times more of it than matter + dark matter.
I think maybe "empty" and "matter" do not mean what you think they mean.
When people tell you that space is empty, what they mean is, there are very few atoms out there. But, that idea of what "empty" means simply does not apply on a subatomic scale. Of course there are very few atoms inside any given atom! One might still argue that there is some kind of "emptiness" inside an atom, but it's a different kind of emptiness. It means something different from the emptiness of space between atoms.
As for "matter," Atoms are not made of matter. Atoms are what matter is made of. Except for mass,* all of the properties of any material substance depend on how its atoms interact with each other. There is no matter (in the usual sense of the word) inside of an atom because there are no interacting atoms inside of an atom. Again, you could still argue that there is some kind of "matter" inside an atom, but it's a different kind of matter from any matter that you can see or touch.
Everything is different at the subatomic scale.
* The mass of any material substance approximately† is the sum of the masses of all of the subatomic particles of which its atoms are made.
† See comment by PM 2Ring, below. You must supply energy in order to separate the components of an atom from each other. Mass is energy. The sum of the masses of the separate components is slightly more than the mass of the atom that they constituted.
This question is kind of tricky. When you go to subatomic distances, "filled" becomes a confusing adjective. In quantum mechanics, objects behave as probability waves. So, they actually fill all the space with some probability. When we say that atoms are mostly empty, this is because we are measuring (like Thomson did in his well-known experiment) the interaction between two particles, so the probability of interaction defines in some way the density of the target that can be seen by our sample particle.
This means that this "density" (named Cross Section to speak more properly) changes with both the target and sample. This is very easy to understand in terms of the macroscopic world. When you hit a wall, it seems to be fully "filled", but an electron or the WiFi signal can go through walls. Even light has a self distance, which is its wavelength, that determines the cross-section of scattering with matter.
How "filled" is the world? Depends on how much resolution you are measuring. We don't know if there is something more small than Quarks. If string theory is correct, all the matter is made of 1D objects, so the volume can't be integrated, and the volume of occupied space can be mathematically zero. This makes no sense because the question has no sense. You are not determining the shape of your sample to measure this density.
