Small non-measurable sets

Question

The standard construction of a subset of [0,1] that is not Lebesgue measurable gives a set with the property that any measurable subset has measure 0. (We might think of these as "small" non-measurable sets.) Suppose two sets (measurable or not) have this property, i.e., that any measurable subset is a set of measure zero. Does the union of these two sets also have this property? One interesting case is when each set, as well as the union, is non-measurable.