You've actually asked a profoundly interesting question for which the proper answer would require a survey of much of current physics. I'll just try to touch on a few points in my reply --- more of an introduction to the questions which your master question implies, rather than the "answers" which are all works in progress.
We know that atoms have mass -- and that some parts of them have electrical charge -- these are all factors observable in bulk and at a distance. From shooting high-speed objects (alpha particles, which are really the nuclei of helium atoms) at an extremely thin gold film, the physicist Ernest Rutherford first established the surprising fact that most of the alpha particles went right through the film, but a remarkable few bounced back right in the direction they had come from. He commented later he was so flabbergasted, it was as if you had shot a bowling ball at at that film and it had come back and hit you.
This observation, and some mathematical analysis of the distribution of scattering of the reflected alpha particles, established the existence of a heavy but quite small nucleus in an atom containing the great majority of an atom's mass.
You might gather from this that "volume", when spoke of at scales this small, requires some careful attention as to the process of measurement. After all, if one were to send very low-speed particles (say, hydrogen atoms) towards a gold film, the majority of these atoms would bounce -- because the primary interaction in this case would be between the electron clouds of the incipient particles and the target.
So the process of defining "volume" is really one of defining distance scales over which certain types of interactions occur. Electrical interactions between electrons, for example, appear to follow the famous inverse square law (that is, if you move two electrons twice as close together, they will repel each other four times as much) -- and so far as we know this behavior continues to incredibly small scales -- this is why electrons are commonly referred to as "pointlike". Protons, however, repel according to the inverse square law only at atom-like distances -- closer in the repulsion goes up much more dramatically as another form of interaction (called the strong force) comes into play. So a proton is said to have a "hard core" towards interaction -- this is essentially saying that it acts like it has "volume".
The point I am trying to make is that volume is not a particularly useful concept at the smallest scales when taken alone -- it is merely a synonym for a variety of different processes of measurement, which on this scale give different results. So instead of referring to "volume" we really must get more specific and describe the behavior of specific interactions which we can measure.
It is commonly said that atoms consist mostly of "empty space" -- i.e. if one applied sufficient pressure to cause electrons to interact with the protons in the associated nuclei and thus convert everything to neutrons, which have no electrical attraction or repulsion, the only remaining repulsion would be that of the "hard core" of the strong force between these particles. And in this case, the "volume", that is, the space which this ensemble would occupy before much more pressure was applied, would be a certain amount, vastly reduced from the size of the original collection of atoms. Thus one could say that if the Empire State Building was crushed down to the point where all the electrons were squeezed into the protons and only neutrons remained, packed at the same pressure which had forced the combination, then this compressed mass would be hardly visible in size. The entire earth subjected to such treatment would only be a few feet across.
But asking what "empty space" is opens another can of worms - the quantum mechanical world created by physicists in the 1920's and 30's to describe the behavior of matter at the smallest scale required the introduction of the idea that space itself is the playground of both the directly perceived real particles and virtual particles which are merely unrealized versions of real particles waiting to hatch, should the energy to promote them to reality be found. One way of visualizing this is to imagine creating a tremendously strong electrical field, a much larger version of what could be generated by connecting two conducting plates to a source of high voltage. In such a field, if one were to place an electron in the middle, it would fall towards the positive plate (a proton would fall towards the negative plate). With sufficiently high field strength, the vacuum itself breaks down - out of the "nothingness", an electron and positron pair (the positron is a sort of "mirror electrical image" of an electron) can pop into being, assume real existence, and each would travel to the correspondingly opposite electrical plate, acquiring the energy for their existence from the high field strength created between the plates.
So there really isn't such a thing as "empty space" -- space itself is a busy substrate which itself seems to play a major role in defining what "particles" are.
I'll conclude by quoting a poem by John Updike describing an author's attempt to come to terms with the confusing interactions (or lack thereof) of one particular particle which does not interact either electrically or with the strong force, and thereby is quite difficult to detect, hardly interacting at all.
Neutrinos, by John Updike
Neutrinos: they are very small
They have no charge; they have no mass;
they do not interact at all.
The Earth is just a silly ball
to them, through which they simply pass
like dustmaids down a drafty hall
or photons through a sheet of glass.
They snub the most exquisite gas,
ignore the most substantial wall,
cold shoulder steel and sounding brass,
insult the stallion in his stall,
and, scorning barriers of class,
infiltrate you and me. Like tall
and painless guillotines they fall
down through our heads into the grass.
At night, they enter at Nepal
and pierce the lover and his lass
from underneath the bed. You call
it wonderful; I call it crass.