That demo of mine cited by amWhy shows that a paradox in the whole hyperbolic plane is possible using pieces that are simple (hyperbolic) triangles. These are indeed measurable sets. That is ok, since the total measure of H^2 is infinite, so it just shows that infinity = 2 infinity. More precisely, it shows that a certain type of measure in H^2 (a type of total measure 1, which DOES exist in R^2) cannot exist. Additionally, it is a nice pictorial representation of the paradox in the abstract group.
The material is discussed in detail in my book The Banach-Tarski Paradox (almost 30 years in print: I am now starting work on a second edition with G. Tomkowicz!) and also in Chapter 19 of my Mathematica in Action (3rd ed.)
And I admit I was very very terse in that demo on the WRI site (from several years ago) not even explaining, as I did above, what the consequences of such a constructive paradox in the hyperbolic plane are. More on the demo: It is really showing the Hausdorff paradox: Each of the three colored sets, orange, blue, and green, are one third of the space, since they are all congruent. But moving our viewpoint a little, one sees that two of the sets are congruent to the third, so each set is also a half of the plane. So 1/2 = 1/3 (if a congruent measure of total measure 1 existed; so such a measure does not exist).