  We explain a general construction of double covers of quadratic degeneracy
loci and Lagrangian intersection loci based on reflexive sheaves. We relate the
double covers of quadratic degeneracy loci to the Stein factorizations of the
relative Hilbert schemes of linear spaces of the corresponding quadric
fibrations. We give a criterion for these double covers to be nonsingular. As
applications of these results, we show that the double covers of the EPW
sextics obtained by our construction give O'Grady's double EPW sextics and that
an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad's EPW cubes.
