How noncompact is the space of Lebesgue measurable functions?

The space in question is the space $\textfrak M$ of Lebesgue measurable subsets of the unit interval equipped with the usual Fr'echet--Nikodym (semi)metric. We show that there exists a sequence of elements of $\textfrak M$ such that their mutual distances are > 1/2. It seems to be an open problem whether "1/2" can be replaced here by a bigger constant C. We show that C must be smaller than 9/14. Moreover, we present a version of the problem in terms of binary codes.