Quasiclassical theory predicts an existence of a sharp energy gap Eg~ħD/L2 in the excitation spectrum of a long diffusive superconductor-normal metal-superconductor (SNS) junction. We show that mesoscopic fluctuations remove the sharp edge of the spectrum, leading to a nonzero density of states (DOS) for all energies. Physically, this effect originates from the quasilocalized states in the normal metal. Technically, we use an extension of Efetov's supermatrix σ model for mixed NS systems. A nonzero DOS at energies E<Eg is provided by the instanton solution with broken supersymmetry.