I need to prove that no Hamel basis can contain a set of positive measure. I already know that a measurable Hamel basis is of Lebesgue zero measure, but I don't how to you use this.
A basis for what? If the space is finite dimensional a Hamel basis is finite so it has measure zero (and every subset is finite so...). If the space is infinite dimensional you'll have to specify what measure you're using.