The production of $Xi(1321)^{-}$ and $overline{Xi}(1321)^{+}$ hyperons in inelastic p+p interactions is studied in a fixed target experiment at a beam momentum of 158 GeV/textit{c}. Double differential distributions in rapidity y and transverse momentum $p_{T}$ are obtained from a sample of 33M inelastic events. They allow to extrapolate the spectra to full phase space and to determine the mean multiplicity of both $Xi^{-}$ and $overline{Xi}^{+}$. The rapidity and transverse momentum spectra are compared to transport model predictions. The $Xi^{-}$ mean multiplicity in inelastic p+p interactions at 158~GeVc is used to quantify the strangeness enhancement in A+A collisions at the same centre-of-mass energy per nucleon pair.