Is the Banach-Tarski paradox consistent with an invariant mass? As a physicist, I try to imagine  that the "sleight of hand" in the theorem is to split a ball with definable volume  into balls with no definable volume and then put them back into a thing with twice the volume. Somehow in the process the information about the original volume was "lost". I am not talking about entropy, just a layman's description. 
1) Is this description accurate? 
2) If this is accurate, can we still consistently define mass in a way that this mass is conserved during the process?(see below for what I mean by this)
Let us imagine a classical Newtonian noiseless world in which bodies can be continuous. Assume there is a sphere that exists since the beginning of the universe. This sphere is charged with 5 different kinds of charges, and these charges are distributed in the same way as the points used in the theorem. Each of these porous subspheres also have mass. This gives us tools to split and reunite the spheres (let's say, by switching on a force in an outside shell that attracts or repulse the sphere rays radially). Is it mathematically consistent a world in which the Banach-Tarski theorem is true, but the mass is conserved?, so that when the spheres are split the mass also splits (and is well defined) and when the spheres are reunited in a different configuration this mass is still conserved (if we now have two spheres the density would be half of the original). That is, can I consistently define mass independently of volume, so that even if we do not have a volume we can still have a mass?   
 A: Too long for a comment. 
Banach-Tarski theorem shows that some conditions imposed on the volume (or mass) cannot hold simultaneously. I think that the answer to your Question 2 depends on which properties mass should have. Do we require that each set have a mass? Is mass translation and rotation invariant? That is, is mass $m(B)$ of a copy $B$ of a set $A$, which is shifted and rotated with respect to the set $A$ the same as mass $m(A)$ of the set $A$? Ignoring mass invariance we can simply define it via delta function, choosing a point $x_0$ and setting $m(A)=1$ if $A$ contains $x_0$ and $m(A)=0$, otherwise. 
Which additivity properties should have a mass? I mean the following. If $A$ is a union of a disjoint  family $\{A_i: i\in I\}$ then $A$ have mass and $m(A)=\sum \{m(A_i): i\in I\}$. Usually it is required that this property takes place for each finite $I$ (additivity) of each countable $I$ ($\sigma$-additivity). I remark that since the group $\Bbb R^3$ is abelian, it is so-called amenable and it admits finitely additive (but not necessarily $\sigma$-additive) translation invariant (but not necessarily rotation invariant) non-trivial mass (that is $m(\Bbb R^3)=1$) which is defined on each subset $A$ of $\Bbb R^3$. 
A demand that mass additivity holds for uncountable cardinality of the set $I$ may lead to the following paradox. Let $A$ be an uncountable set. If uncountably many points of $A$ has positive mass, how we sum these values to obtain mass of $A$? Conversely, if each point of $A$ has mass $0$, then how can mass of $A$ be non-zero? Even mass $\sigma$-additivity with its translation invariance can lead to contradictions, similarly to that used in a construction of a non-measurable Vitali set.
The above paradoxes may lead to a conclusion that the mass is a property not of each set and it is not concentrated in points, similarly to other physical notions. I recall Zeno “arrow” aporia on motion. When a flying arrow moves if at each moment it has a fixed position? Also how time can have duration if it is composed of moments none of which has duration? 
