The measurement of the central exclusive production of charged hadron pairs $h^{+}h^{-}$ ($h = \pi, K, p$) by the STAR experiment at RHIC is reported. The data from proton-proton collisions at $\sqrt{s} = 200$~GeV were used in this study. The pairs of charged hadrons produced in the reaction $pp\to p^\prime+h^{+}h^{-}+p^\prime$ were reconstructed from the tracks in the central detector, while the forward-scattered protons were measured in the Roman Pot system. Differential cross sections were measured in the fiducial region determined by the geometrical acceptance of the experimental setup. They were compared to phenomenological predictions based on the Double Pomeron Exchange model. The fiducial $\pi^+\pi^-$ cross section was extrapolated to the Lorentz-invariant region, which allowed decomposition of the invariant mass spectrum into continuum and resonant contributions.