Banach-Tarski Paradox for the unit interval?

One cannot expect an explicitly constructible example. By an old result of Solovay, if ZF (Zermelo-Fraenkel Set Theory without Choice) plus "there exists an inaccessible cardinal" is consistent (which is generally believed to be the case), then so is ZF plus "every set of reals is Lebesgue measurable."

The sets involved in the paradoxical decomposition cannot all be Lebesgue measurable. Indeed thinking of them as measurable is what makes the decomposition seem paradoxical. Thus some version of the Axiom of Choice is needed, and an explicit set-theoretic construction from just ZF is not possible.

As for "one dimensional" paradoxical decompositions into a finite number of sets, these are not possible. For it can be shown that there is a finitely additive translation-invariant "measure" that extends Lebesgue measure and is defined on all sets of reals. Using such a "measure" we can translate the intuitive notion of paradoxicalness into a proof that there is no paradoxical decomposition. The Banach-Tarski Paradox can be thought of as a proof that there is no translation invariant finitely additive "measure" defined on all subsets of $\mathbb{R}^3$.

To get such a paradox using finitely many pieces, as in the Banach-Tarski case, one must move beyond isometries. But if one uses measure-preserving transformations, such a paradox does exist on the line. Reviewing my book on the BTP I find Theorem 7.9 which shows that there is a paradoxical decomposition of the interval using transformations preserve Lebesgue measure. The main point is that there is a Lebesgue-measure preserving function from the interval to the sphere, and that can be used to move the Banach-Tarski Paradox down to the line.

And there is von Neumann's paradox (Thm 7.12) which provides a finitary paradoxical decomposition of the line using epsilon contractions, for any positive epsilon. Since such a contraction is a linear fractional transformation it shrinks Lebesgue measure, so this can be considered paradoxical. Both of these use the Axiom of Choice.

Stan Wagon

Actually, let's work with the circle ${\mathbb T} = {\mathbb R}/{\mathbb Z}$, and $\mathbb Q_T = {\mathbb Q}/{\mathbb Z}$. A Vitali set $V$ is a subset of $\mathbb T$ such that $\mathbb T$ is the disjoint union of the translates $V + r$ for $r \in \mathbb Q_T$. Of course this can't be constructed explicitly, you need the Axiom of Choice. Now for $r \in [0,1/2)/\mathbb Z$ translate $V + r$ to $V + 2r$, and the disjoint union of these is $\mathbb T$; for $r \in [1/2, 1)/\mathbb Z$ translate $V + r$ to $V + 2r - 1$, and the disjoint union of these is also $\mathbb T$. So we've decomposed $\mathbb T$ into countably many pieces, translated some of them to make $\mathbb T$, and translated the rest to make another copy of $\mathbb T$.

If you take $[0,1)$ instead of $\mathbb T$, you can make this work with a slight adjustment: instead of translating all of $V + r$ by $r$ to get $V+2r$, you have to break $V+r$ into two pieces, translate $(V + r) \cap [0,1-r)$ by $r$ but translate $(V + r) \cap [1-r, 1)$ by $r-1$.

If you want to use $[0,1]$ instead of $[0,1)$ there's a further complication: what do you do with $1$?. But that's not hard: noting that all the translations above were by rational numbers, you just do what I described to the irrationals in $[0,1]$, and separately map the rationals in $[0,1]$ to two copies of itself.