In 1920 H. Steinhaus [Sur les distances des points de mesure positive, Fundamenta Mathematicae 1 (1920) 93-104.] proved the following result: ``Let A be a Lebesgue measurable set of positive measure. Then there exist at least two points in $A$ such that the distance between them is a rational number''. In this paper we shall prove that there exists a sequence $(x_n)n\geq1$ of different points in $A$ such that the distance between any two of them is a rational number. Further, we shall extend our result to the case when $A$ is a set with the Baire property (non-necessarily Lebesgue measurable).