We construct a systematic low-temperature theory to calculate smectic elastic constants of layered phases of self-avoiding fluid membranes. At the lowest, harmonic level, this theory reproduces the results of Helfrich's theory of entropically generated steric intermembrane interactions. At the lowest order beyond the harmonic approximation, our theory includes the influence of crumpling of individual membranes on the smectic elasticity of lamellar fluid membrane phases. We discuss in detail crumpling corrections to Helfrich's expressions for the smectic bending and compressibility elastic constants ${K}_{\mathrm{sm}}$ and ${B}_{\mathrm{sm}}$, as well as to the exponent \ensuremath{\eta} characterizing the decay of smectic density correlations and to the de Gennes penetration depth \ensuremath{\lambda}, both of which have been measured in recent scattering experiments on lamellar multimembrane phases. We argue that the asymptotic value of \ensuremath{\eta} is nonuniversal in the limit in which the repeat distance l of the layering is much larger than the membrane thickness.Though this exponent does not exhibit (at the lowest level beyond harmonic approximation) logarithmic l-dependent corrections (which are, however, present in ${K}_{\mathrm{sm}}$, ${B}_{\mathrm{sm}}$, and \ensuremath{\lambda}), it does exhibit some temperature dependence. We demonstrate that lamellar phases are stabilized by an interplay betwween suitably defined, negative surface tension and the self-avoidance of neighboring membranes. Throughout the paper we try to clarify the role of surface tension in lamellar systems and in fluid membrane systems in general. In particular, we show that the presence of surface tension in lamellar multimembrane phases is neither inconsistent with nor violates the Landau--Peierls--de Gennes elastic free energy for a layered solid. For this purpose we construct several thermodynamic ensembles to describe fluctuating fluid membranes. The bulk of our calculations is done in a constant-area ensemble. Our results, concerning crumpling effects on the softening of fluid membrane rigidity, are in quantitative agreement with other low-temperature theories and in disagreement with Helfrich's prediction. However, we believe that our approach indicates possible reasons for the breakdown of Helfrich's qualitative arguments.