Do subatomic particles have finite (i.e. non-zero) size? We know atoms are mostly "made" out of empty space, so the nucleus and all the subatomic particle are very small in compared to the magnitude of the atoms. We also know that atoms are incredibly small, so how small are subatomic particle? Can we measure their size?
Is it meaningful to even ask this question?
 A: In physics, fundamental particles are typically treated as point particles. In this approximation, they have no size or shape whatsoever. They sort of have a location, but we can never exactly pinpoint this location in space, because quantum mechanics tells us that a particle never has an exact location.
The classical model of the electron does yield a non-zero radius, but this model is completely inaccurate when describing the size of the electron. It should not be used in any dimension-related context.
Protons and neutrons (and hadrons) are composite particles, though still subatomic. They do not have a point mass because they are made up of quarks and gluons.
A: In the relativistic "Einstein-Cartan Theory", observable fermions have some spatial extent greater than the Planck length, although it would still be incredibly tiny. ECT was developed through conversations between Einstein and the mathematician Elie Cartan, five years after the discovery of particulate spin in the mid-1920's.
It matches all observational predictions of 1915's General Relativity, although it remains much less well-known than GR, partly because its math is much more complicated, and partly because it missed the great boom in GR that had been sparked by the Eddington team's well-publicized observations of the 1919 eclipse from vantage points both in South America and in Africa; also, Pauli's mid-1920's concern that fermionic spin would exceed "the speed of light in vacuum" seemed (for superposition of matter and energy) to require a treatment of fermions as "point-like".
As Pauli was dealing with subatomic quanta, rather than cosmology, he disregarded the clear fact that the speed of light in vacuum could vary locally between causally-separated regions (i.e., black holes, which had been hypothesized by Laplace and other scientists at least as early as 1793, but for which there was no astronomical evidence before the 1930's).  Einstein described the possibility of such variations verbally and explicitly (but not mathematically), in "popularized" versions of his theory published as early as 1916.
As described between 2009 and 2020 in numerous papers by Nikodem J. Poplawski that are available free on the Arxiv website, ECT allows a relatively simple cosmology.  In his model, the gravitational collapse of any large rotating star, occurring after the depletion of its nuclear fuel would have eliminated the radiation pressure that might have otherwise prevented that collapse, would put fermions newly-materialized from matter fields (through the separation of virtual pairs of widely-spaced fermions by the "event horizon" propagating outward from the collapsing star's center) into contact with the vastly larger stellar fermions, which would remain larger by 32 orders of magnitude.  Because all fermions spin, that contact would reverse and greatly accelerate the trajectories of many of the newly-materialized ones, which would form a new "local universe" (marginally within the outer portion of the volume that had been occupied by the star, but on smaller spatial and temporal scales), where the older section of their trajectories would remain visible as its future inhabitants' "Cosmic Microwave Background" radiation.
Because of the causal separation between the collapsed star's maximum volume and the spacetime of the "parent" universe (which is left inaccessible from within that volume, due to the limited amount of energy remaining within it), the outward motion of the newly-materialized particles is identical to that simulation of their motion which is more poetically described as an effect of the relativistic expansion of space (in an analogy dating more directly from that static universe which Einstein had envisaged in 1915) than from a dynamical universe whose gravitational field may be more comparable to the electromagnetic and nuclear fields of quantum physics.  (Guth, for instance, often compares gravity to "negative energy".)
The resemblance between the ECT and GR models might presumably be traced to Einstein's generalization of the concept of fields, as their effects on the observed CMB have been shown by Desai to be identical, at
https://arxiv.org/abs/1510.08834 .
As the modeled progenitor (possibly eternally old) of an infinity of local universes on sequentially-decreasing scales, Poplawski's model is often described as an inflationary one, although the mechanism of that result does not rely on the existence of the scalar field of earlier models, whose hypothetical "inflaton" particles have not proven to be observable.  (Essentially, the quasi-exponential expansion of space would still happen, but would result from its expansion into volumes previously occupied by particles motivated by positive or negative versions of fields already known, rather than from any overall "shape".)
A: I'm not sure that the (more sophisticated) answers presented here are exactly what you are after when you ask:

Can we measure their size?

And

Is it meaningful to even ask this question?

Yes and no..may be an answer to both of these questions.
There may not be a physical size, in the sense of a snooker ball shrunk down to the smallest scales. Nobody can tell you with total confidence the answer to that question, as we are limited by our technology in our ability to check the "sizes" predicted by  various theories.
Practically speaking however, the physical "size" of any object at this scale is not very important and it's not really meaningful to ask how small they are.
There is however, an "effective" size, that is dependent on what property of the particle you want to measure and how accurately we are able to measure it.
This effective size may not be dependent on the physical size of the particle. An example of this is the proton, which although it is although much more massive than the electron, (and composed of 3 quarks), still has the same amount of electric charge as the electron, although defined as positive rather than negative. So physical  size does not matter here.
It's the strength of the charge(s) possessed by the particles that is all we can measure. These charges include the electric charge, but also mass, weak force and strong force and other properties of the particle.
This effective size is important when we want to check our theories of how particles interact with experimental results.
All we can ever measure is this effective size, as to measure the properties of elementary particles we always have to interact with them, so their physical sizes, if they have any in the classical sense,  don't really come into it. 
A: Your question is erroneous, there are no "particles", maybe better, "Do subatomic waves have finite size?".
Subatomic matter is waves (information) on "______?", with no one point location, they have a length location (anywhere along the wave). 
The answer is, yes, everything has a finite size. Mostly likely measured in integer planck lengths (1 planck = ‎1.616×10−35 of a metre).
Space isn't empty, if it was it couldn't be distorted by gravity.   
