C$_2$H is one of the first radicals to be detected in the interstellar medium. Its higher rotational transitions have recently become available with the Herschel Space Observatory. We aim to constrain the physical parameters of the C$_2$H emitting gas toward the Orion Bar. We analyse the C$_2$H line intensities measured toward the Orion Bar CO$^+$ Peak and Herschel/HIFI maps of C$_2$H, CH, and HCO$^+$, and a NANTEN map of [CI]. We interpret the observed C$_2$H emission using radiative transfer and PDR models. Five rotational transitions of C$_2$H have been detected in the HIFI frequency range toward the CO$^+$ peak. A single component rotational diagram gives a rotation temperature of ~64 K and a beam-averaged C$_2$H column density of 4$\times$10$^{13}$ cm$^{-2}$. The measured transitions cannot be explained by any single parameter model. According to a non-LTE model, most of the C$_2$H column density produces the lower-$N$ C$_2$H transitions and traces a warm ($T_{\rm{kin}}$ ~ 100-150 K) and dense ($n$(H$_2$)~10$^5$-10$^6$ cm$^{-3}$) gas. A small fraction of the C$_2$H column density is required to reproduce the intensity of the highest-$N$ transitions ($N$=9-8 and N=10-9) originating from a high density ($n$(H$_2$)~5$\times$10$^6$ cm$^{-3}$) hot ($T_{\rm{kin}}$ ~ 400 K) gas. The total beam-averaged C$_2$H column density in the model is 10$^{14}$ cm$^{-2}$. Both the non-LTE radiative transfer model and a simple PDR model representing the Orion Bar with a plane-parallel slab of gas and dust suggest, that C$_2$H cannot be described by a single pressure component, unlike the reactive ion CH$^+$, which was previously analysed toward the Orion Bar CO$^+$ peak. The physical parameters traced by the higher rotational transitions ($N$=6-5,.,10-9) of C$_2$H may be consistent with the edges of dense clumps exposed to UV radiation near the ionization front of the Orion Bar. ; Comment: Proposed for acceptance in A&A, abstract abridged