Nonmeasurable set
From Academic Kids

In mathematics, a nonmeasurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume.
This page gives a nontechnical description of this concept. For a technical description see measure (mathematics) and the various constructions of nonmeasurable sets, Vitali set, Hausdorff paradox, BanachTarski paradox.
The first indication that there may be a problem to define length for any set came from Vitali's theorem which basically states that you can take an interval of length 1, dissect it into pieces, move the pieces around and get an interval of length 2 (sometimes this result is called the Hausdorff paradox). However, it is necessary that the number of pieces be infinite. Hence one could possibly interpret this result as saying that the correct length of each of these pieces is 0, but when you add them up you could get 1 or 2. Such a definition of length is called a finitely additive measure.
When you increase in dimension the picture gets worse. The BanachTarski paradox claims that you can take a ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1. Obviously this construction has no meaning in the physical world. In 1989, A. K. Dewdney published a letter from his friend Arlo Lipof in the computer recreations column of the Scientific American where he describes an underground operation "in a South American country" of doubling gold balls using the BanachTarski paradox. Naturally, this was in the April issue.
Thus there is no way to define volume in three dimensions unless one of the following four concessions are made
 The volume of a set might change when it is rotated
 The volume of the union of two disjoint sets might be different than the sum of their volumes
 Some sets might be tagged "non measurable" and one would need to check if a set is "measurable" before talking about its volume
 The rules of mathematics might be altered to disallow the above constructions.
It turns out that the price to pay for concession 3 is surprisingly small. The family of measurable sets is very rich, and almost any set you run into in most branches of mathematics is measurable. It is not possible to construct a nonmeasurable set, only to show indirectly that one exists. It is also typically very easy to prove that a given specific set is measurable. Therefore this is the preferred option for most mathematicians. As an added bonus, you get that even an infinite series of disjoint sets satisfies the sum formula, a property mathematicians call σadditivity.
On the other hand, the price for concession 4 is also smaller than one would expect. It turns out that a specific axiom can be pointed out as being at fault. This is the famous axiom of choice. It turns out that removing this axiom from mathematics only changes small, easy to identify areas, and the bulk of mathematics stays unaltered. See axiom of choice for a full discussion. This is mathematicians' second most popular choice.
Finally, the notion that giving away σadditivity in one dimension to get a definition of length for all sets has not proved to be very useful. A very short discussion of the reasons can be found in measure (mathematics).
References
 A. K. Dewdney, A matter fabricator provides matter for thought, Computer recreations, Sci. Am. April 1989, 116119.