We simulate a hard-sphere liquid in confined geometry where the separation of the two parallel, hard walls is smaller than two particle diameters. By systematically reducing the wall separation we analyze the behavior of structural and thermodynamic properties, such as inhomogeneous density profiles, structure factors, and compressibilities when approaching the two-dimensional limit. In agreement with asymptotic predictions, we find for quasi-two-dimensional fluids that the density profile becomes parabolic and the structure factor converges toward its two-dimensional counterpart. To extract the compressibility in polydisperse samples a perturbative expression is used which qualitatively influences the observed nonmonotonic dependence of the compressibility with wall separation. We also present theoretical calculations based on fundamental-measure theory and integral-equation theory, which are in very good agreement with the simulation results.