The quantum mechanical Beth-Uhlenbeck (BU) approach for repulsive hard-core interactions between baryons is applied to the thermodynamics of a hadron gas. The second virial coefficient $a_2$ – the ``excluded volume’’ parameter – calculated within the BU approach is found to be temperature dependent, and it differs dramatically from the classical excluded volume (EV) model result. At temperatures $T =100-200$ MeV, the widely used classical EV model underestimates the EV parameter for nucleons at a given value of the nucleon hard-core radius by large factors of 3-4. Previous studies, which employed the hard-core radii of hadrons as an input into the classical EV model, have to be re-evaluated using the appropriately rescaled EV parameters. The BU approach is used to model the repulsive baryonic interactions in the hadron resonance gas (HRG) model. Lattice data for the second and fourth order net baryon susceptibilities are described fairly well when the temperature dependent BU baryonic excluded volume parameter corresponds to nucleon hard-core radii of $r_c = 0.25-0.3$ fm. Role of the attractive baryonic interactions is also considered. It is argued that HRG model with a constant baryon-baryon EV parameter $v_{NN} \simeq 1$ fm$^3$ provides a simple yet efficient description of baryon-baryon interaction in the crossover temperature region.