Where do the mass and volume of the two quarks go when they create a meson? A quark and an antiquark are fermionic particles with mass. Where do this said mass go when they bind to create a meson?
If there's annihilation at play, how do mesons even be in the first place?
And lastly, how exactly does volume in a meson, a bosonic particle made up of fermionic standard particles, 'work'? Mesons by definition should not be obeying Pauli's exclusion principle. What happens to the quarks' volume then?
A semi-detailed answer which a beginner can understand would be really appreciated.
 A: 
A quark and an antiquark are fermionic particles with mass. Where do this said mass go when they bind to create a meson?

The mass of the quarks contributes to the mass of the meson. However, this is typically a small contribution. The mass of a meson is typically much larger than the sum of the masses of the constituent quarks; the remaining mass comes from the energy of the strong interaction between the quarks.

If there's annihilation at play, how do mesons even be in the first place?

Annihilation exists, but it doesn't have to happen immediately. For example, a bound state of an electron and a positron, called positronium, can exist for quite some time before the electron and positron annihilate. The same is true of quarks and antiquarks.

And lastly, how exactly does volume in a meson, a bosonic particle made up of fermionic standart particles, 'work'? Mesons by definition should not be obeying Pauli exclusion. What happens to the quarks' volume then?

Quarks are point particles, so they don't have any volume. Composite bosons made of fermions are common in other areas of physics (for example, Bose-Einstein condensates work in a similar way), so there's nothing particularly special about mesons in this respect.
A: When a quark and an antiquark bind into a meson, they don’t lose their mass. Their mass contributes to the mass of the meson, along with their kinetic energy and their potential energy. There is potential energy associated with the strong nuclear force between them and also potential energy associated with the electromagnetic force between them.
Such mesons decay after a short time because the quark and antiquark annihilate and their energy forms other particles.
As far as we can tell, quarks are point particles and have no volume. But the meson that is their bound state has a size for the same reason that a hydrogen atom has a size.
A: The other answers address the mass, but none of them really gives you a real reason for the volume, so I will address that in detail.
Quarks and antiquarks are elementary particles, pointlike, with no spatial extension or substructure as per the SM.
Mesons, are hadronic subatomic particles made up of a quark and antiquark pair, bound by the strong force.

Because mesons are composed of quark subparticles, they have physical size[further explanation needed], notably a diameter of roughly one femtometer,1 which is about 1.2 times the size of a proton or neutron.

https://en.wikipedia.org/wiki/Meson
Now you are asking, because the PEP does not apply to bosons, why does the composite meson have a volume? 
Now quarks are fermions, and they do obey the PEP. But it is not that simple. The PEP is only for identical fermions. The quark and antiquark cannot come close together (more precisely why they cannot occupy the same QM state) because of the PEP is not correct to say (since they are not identical). There are more reasons why, and one of them is because at very short distances there is the phenomenon of asymptotic freedom, which will cause the binding force to become less important.

There are actually two more features in the soup that might address the OP's question. First, as two quarks get far too close, asymptotic freedom makes their gluon interaction essentially insignificant -- their are freed from each other. And if they are the same kind they exclude each other by the Pauli principle; if not some type of antisymmetrization may also be provided by a generalized version of that principle.

https://physics.stackexchange.com/a/396041/132371
The other reason is why the composite meson (or anything composite made up of quarks) does have a volume, is that you have a false imagination of how quarks make up mesons. You believe mesons are just a quark and antiquark. In reality, it is called a sea of quarks, antiquarks, and gluons, that creates the meson, and only if you take a net picture of the meson (considering conservation laws and annihilation), then will you find a quark and a antiquark left.

Quarks and antiquarks and gluons dance around and annihilate and pair produce in a non stop manner, so they do "overlap" in the feynman diagrams of the individual interactions, and annihilate. The three valence quarks are lost in the soup, and in any case it is a matter of conservation of quantum numbers, there should be an excess of one down and two up for the proton.
  So it is not a matter of repelling, it is just that overall the quarkness up and down should add up to the valence quarks of a proton, and the same holds for the neutron two down one up excess in the soup .


https://physics.stackexchange.com/a/396041/132371
A: To get a feeling how strong interactions tie up hadrons, look at this illustration of a proton, which is a bound state by three quarks, called valence , counting quantum numbers.

Something similar could be drawn for a pion.
Because strong interactions are strong,  the coupling constant is 1, an innumerable number of gluons and antiquark quark pairs can be supposed to exist within a hadron, continually created by gluons and annihilated into gluons.  The sum of the  quantum numbers adds up to the quantum numbers of the valence quarks which characterize the hadron.
Fortunately we are in the special relativity regime, which means that the four vectors of all this mess add up to  a four vector whose invariant mass is the mass of the hadron, much larger than the mass of the valence quarks. The valence quarks are  below 5 Mev, and the pion over 100.  In any case the masses within the hadron bag are off mass shell, it is the summed four vector mass that is important.
All the above is qualitative, because no calculations can be done with usual Feynman diagrams for strong interactions. New tools have been found and are being sought. QCD on the lattice has been fairly successful in calculating the mass spectra of hadrons, and research is still going on. 
As for the Pauli exclusion principle , there are so many energy levels in this mess, that there is no difficulty for all these fermions to exist in one of them, again talking qualitatively, as it is  not possible to calculate them.
