We present two successive mean-field approximations for describing the mechanical properties and the swelling equilibrium of polyelectrolyte gels in contact with a salt solution. The first mean-field approximation reduces the many-chain problem of a gel to a corresponding single-chain problem. The second mean-field step integrates out the degrees of freedom of the flexible chain and the ions. It replaces the particle-based description of the polyelectrolyte with suitable charge distributions and an effective elasticity term. These simplifications result in a computationally very efficient Poisson-Boltzmann cell-gel description. Despite their simplicity, the single-chain cell-gel model shows excellent and the PB model very good agreement with explicit molecular dynamics simulations of the reference periodic monodisperse network model for varying chain lengths, polymer charge fractions, and external reservoir salt concentrations. Comparisons of our models to the Katchalsky model reveal that our approach is superior for strongly charged chains and can also predict the bulk moduli more accurately. We further discuss chain length polydispersity effects, investigate changes in the solvent permittivity, and demonstrate the robustness of our approach to parameter variations coming from several modeling assumptions.