Translates of the Vitali set are disjoint from it.

Suppose it had a measurable subset of nonzero measure. By translating this subset by all rationals in the unit interval, you obtain a countable collection of disjoint sets of real numbers in the interval [0,2], who all have the same nonzero measure (by translation invariance of measure); their countable union is measurable, and since it's a subset of [0,2], it must have finite measure. See what you can make of all this.