The formalism to augment the classical models of equation of state for real gases with the quantum statistical effects is presented. It allows an arbitrary excluded volume procedure to model repulsive interactions, and an arbitrary density-dependent mean field to model attractive interactions. Variations on the excluded volume mechanism include van der Waals (VDW) and Carnahan-Starling models, while the mean fields are based on VDW, Redlich-Kwong-Soave, Peng-Robinson, and Clausius equations of state. The VDW parameters of the nucleon-nucleon interaction are fitted in each model to the properties of the ground state of nuclear matter, and the following range of values is obtained: $a = 330 - 430$ MeV fm$^3$ and $b = 2.5 - 4.4$ fm$^3$. In the context of the excluded-volume approach, the fits to the nuclear ground state disfavor the values of the effective hard-core radius of a nucleon significantly smaller than $0.5$ fm, at least for the nuclear matter region of the phase diagram. Modifications to the standard VDW repulsion and attraction terms allow to improve significantly the value of the nuclear incompressibility factor $K_0$, bringing it closer to empirical estimates. The generalization to include the baryon-baryon interactions into the hadron resonance gas model is performed. The behavior of the baryon-related lattice QCD observables at zero chemical potential is shown to be strongly correlated to the nuclear matter properties: an improved description of the nuclear incompressibility also yields an improved description of the lattice data at $\mu = 0$.