We extend the statistical bootstrap model (SBM) in order to describe hadronic systems which carry strangeness. We establish that the hadronic phase can exist only in a region of the phase space (T, , ), which is bounded by a `critical' surface. The SBM leads to a mass spectrum with the asymptotic form , where the exponent can take different values. We first employ the usual version of the SBM and impose strangeness neutrality in order to obtain the relation between the temperature and the chemical potentials , . We find an unphysical behaviour of near the critical surface. Subsequently, we study another version of SBM where and we are led to a realistic as well as acceptable behaviour. In this case we are also able to relate the parameter of the model, , (the critical temperature at zero ) with the MIT bag constant B. The relation between (T, , ) can be used to predict the particle ratios on the critical surface at the limit of the hadron gas phase.