Closed sets with the Kakeya property

We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the nontrivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.