Of the numerous mechanisms that have been postulated to explain the origin of biological homochirality, asymmetric autocatalysis coupled with mutual inhibition is often cited as a plausible route to abiotic symmetry breaking. However, in a system closed to mass flow, the constraint of microscopic reversibility ensures that this far-from-equilibrium phenomenon can at best provide a temporary excursion from racemic equilibrium. Comparatively little attention has been paid in the literature to the manner in which such a closed system approaches equilibrium, examining the mechanisms and time scales involved in its transit. We use an elementary lattice model with molecular degrees of freedom, and satisfying microscopic reversibility, to investigate the temporal evolution of stochastic symmetry breaking in a closed system. Numerical investigation of the model's behavior identified conditions under which the system's evolution toward racemic equilibrium becomes extremely slow, allowing for long-time persistence of a symmetry-broken state. Strong mutual inhibition between enantiomers facilitates a "monomer purification" mechanism, in which molecules of the minor enantiomer are rapidly sequestered and a nearly homochiral state persists for long times, even in the presence of significant reverse reaction rates. Simple order of magnitude estimates show that with reasonable physical parameters a symmetry-broken state could persist over geologically relevant time scales.