Is the Banach-Tarski paradox realistic? Why is Volume not an invariant?

The main crux of the Banach-Tarski construction is that you can break up a measurable set into non-measurable sets. Measure is, in a certain sense, analogous to volume. Non-measurable sets are somewhat strange and it can be shown that under mild axiomatic assumptions that it is consistent that there are no non-measurable sets. So in that sense, there is a "reality" where the Banach-Tarski paradox doesn't exist.

Indeed, the Banach–Tarski theorem is not "realistic" in the sense of applying to the physical world. However, I think the physical reason for this that you gave is not adequate, because there is no physical law of conservation of volume. Even without a chemical reaction, volume can change under pressure (especially for gases, but also a little bit for liquids and solids.)

The usual argument against the possibility of a physical realization of the Banach–Tarski theorem is based on the physical law of conservation of mass, not volume. However, I think that argument is also flawed. According to the principle of mass–energy equivalence, the process of cutting up a physical object and separating its pieces adds mass to the system if the pieces are attracted to one another (which is often the case with physical objects.) The reason is that the separation requires energy, which is equivalent to mass. See the article on binding energy for a more thorough discussion of this topic.

(Of course, the resemblance between the physical possibility of increasing the mass of a system of bound particles in this way and the mathematical possibility of duplicating a ball using the Banach–Tarski theorem is purely superficial.)

We shouldn't expect the Banach–Tarski theorem to apply to physical objects, simply because it makes no claim to apply to physical objects. It is about the notion of set, which is an abstract notion separate from the realm of physical objects. One can make limited analogies between sets and physical objects, and the Banach–Tarski theorem is one example among many examples of limitations to these analogies.

The balls in question are definitely not made out of glass or any other material. Even if we were able to break them down into quarks and had the ability to reassemble them at will, we still won't be able to do this. It is misleading to think of the Banach-Tarski paradox in those terms.

The paradox addresses aspects of the usual formalisation of the continuum that don't fit very well with our physical intuition. In this sense, the Banach-Tarski paradox is a comment on the shortcomings of our mathematical formalism. However, once one stops thinking of the oh!-so-real numbers as describing reality in any sense, the paradox ceases being one and becomes an intriguing mathematical result.

When properly channeled, nonmeasurable sets can become a useful tool. For example the closely related notion of an ultrafilter is a powerful mathematical tool.

It is not physically possible to demonstrate the Banach–Tarski paradox. The sets in question are very bizarre and can be best described as a distribution of points. A good example which is related, and is easily understandable is the Vitali set. There is a Wikipedia article for this.

I would like to add only a little to what has been said. User2566092 points out the unmeasurability of the sets we use. The fact that non measurable sets exists should give one pause for thought on its own I think.

The problems inherent in trying to cut a sphere like that in real life are multiple. For one the sets you are using are very much scattered. They are the representatives of orbits under a relation after all. Second to get the representatives we need to use the Axiom of Choice. That again shows something is not going quite right. If Axiom of Choice is necessary then we are not only choosing from too many sets already, those sets also pretty much have to be quite strange, for example you can't get a reasonable (definable without AC) well order on them.

Now before I give the impression that if we just got rid of AC all would be well in the mathematical world, let me point out that many (most) of the alternatives are often quite awful as well. For example if you want measurability of all sets you might be tempted to go with AD (axiom of determinacy) which seems nice you get all subsets of the reals are Lebesgue measurable, have the property of Baire, and the perfect set property. Unfortunately you also get consistency of ZF. Which is bad enough, but worse yet you actually get inner models some "very" large cardinals. Now this (to me) says that AD is in some sense much more insane then AC. You even get that some cardinals which are usually very large (measurable) are actually quite small $\omega_1$. You might want to take a look at https://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy which is were I cribbed some of these results from.

What I'm trying to say is, although AC gives us some pretty strange beasts. Its main rival doesn't do much better. If you just drop AC altogether you lose lots of things you really tend to want. Like the fact that sets have cardinalities and are comparable.

So really once you go past "small" finite sets you tend to get many strange results.