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Context. Quasi-periodic X-ray eruptions (QPEs) are a recently discovered phenomenon, the nature of which remains unclear. Based on their discovery in active galactic nuclei (AGNs), explanations related to an AGN accretion disk or potentially a stellar tidal disruption event (TDE) have been put forward. Alternatives, including highly unequal mass compact object binaries, have also been proposed to explain their properties. Aims. We perform a systematic study of the five known QPE host galaxies with the aim of providing new insights as to their nature. Methods. We analysed new and archival medium resolution optical spectroscopy of the QPE hosts. We measured emission (and absorption) line fluxes, their ratios, and equivalent widths (EWs) to locate the QPE hosts on diagnostic diagrams. We also measured the velocity dispersion of the stellar absorption lines to estimate their black hole masses. Results. All QPE host galaxies show emission lines in their optical spectra. Based on their ratios and EWs, we find evidence for the presence of an AGN in all sources, including those previously reported as passive. We measure velocity dispersions between 36 and 90 km s −1 , implying the presence of low mass (10 5−6.7 M ⊙ ) black holes, consistent with literature findings. Finally, we find a significant over-representation (two out of the five sources, or a factor of 13 −10 +13 ) of quiescent Balmer strong (post-starburst) galaxies among QPE hosts. Conclusions. The presence of a narrow line region consistent with an AGN in all QPE host galaxies implies that a long-lived accretion flow likely plays an integral part in the QPE phenomenon. The strong over-representation of quiescent Balmer strong galaxies among QPE hosts can be naturally explained in both the TDE and interacting extreme mass ratio inspiral hypotheses.