Vitali set
A Vitali set is the original, classic example of a set that cannot be measured. It is built by a single, decisive use of the axiom of choice: you sort the points of [0, 1] into families that differ only by a rational shift, then choose exactly one representative from each family. The resulting set is so irregular that assigning it any length collapses into contradiction.
Construct it precisely as follows. Declare two reals equivalent if their difference is rational; this is an equivalence relation, partitioning [0, 1] into uncountably many classes. By the axiom of choice, pick one representative from each class to form a set V. Now consider the countably many translates V + q for rationals q in [-1, 1]; these translates are pairwise disjoint, and their union contains [0, 1] while sitting inside [-1, 2].
If V were measurable with measure m(V), then by translation invariance every translate has the same measure m(V), and countable additivity gives the total measure as the sum of countably many copies of m(V). That sum is 0 if m(V) = 0 and +infinity if m(V) is positive — but it must lie between 1 and 3, the measures of [0, 1] and [-1, 2]. Both cases are impossible, so V is non-measurable. The argument is a clean clash between translation invariance and countable additivity.
The crux in numbers: the disjoint translates V + q satisfy 1 = m([0,1]) is at most sum of m(V+q) is at most m([-1,2]) = 3. Each m(V+q) equals the single number m(V), so the middle sum is 0 (if m(V)=0) or +infinity (if m(V)>0). Neither lands in [1, 3]; contradiction.
Translation invariance plus countable additivity force an impossible total.