We consider a four-field formulation of Biot's quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy's law results in a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces. Next, we propose a family of H(div div)-H(div)-conforming mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish a priori error estimates for the related family of fully conservative finite element methods.