The shear viscosity $eta$ in the van der Waals excluded volume hadron-resonance gas model is considered. For the shear viscosity the result of the non-relativistic gas of hard-core particles is extended to the mixture of particles with different masses, but equal values of hard-core radius r. The relativistic corrections to hadron average momenta in thermal equilibrium are also taken into account. The ratio of the viscosity $eta$ to the entropy density s is studied. It monotonously decreases along the chemical freeze-out line in nucleus-nucleus collisions with increasing collision energy. As a function of hard-core radius r, a broad minimum of the ratio $eta/sapprox 0.3$ near $r approx 0.5$ fm is found at high collision energies. For the charge-neutral system at $T=T_c=180$ MeV, a minimum of the ratio $eta/scong 0.24$ is reached for $rcong 0.53$ fm. To justify a hydrodynamic approach to nucleus-nucleus collisions within the hadron phase the restriction from below, $r~ ge ~0.2$ fm, on the hard-core hadron radius should be fulfilled in the excluded volume hadron-resonance gas.
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