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Context. The X-ray output of low-metallicity starburst galaxies is a key component of stellar feedback, tracing the processes responsible for gas ionization and chemical enrichment. The integrated X-ray luminosity ( L X ) from high-mass X-ray binaries in star-forming galaxies scales with the star formation rate (SFR) and the host galaxy metallicity Z . Due to the inverse correlation between L X /SFR and Z , the contribution of X-ray binaries to the ionizing photon budget is expected to be enhanced in metal-poor systems. Their radiation can potentially ionize He II in the surrounding interstellar medium, powering nebular He II λ 4686 Å emission. However, detailed studies of the X-ray emission in individual low- Z starburst galaxies are rare, and their X-ray properties are not well explored. Aims. The blue compact dwarf galaxy ESO 338-IG04 (ESO 338-4 hereafter) serves as a nearby template for studying stellar feedback and X-ray emission in low-metallicity starbursts. It combines vigorous recent star formation, a low metallicity (12 + log(O/H)≈7.9, or 12% solar), and a rich population of massive stellar clusters. Extensively observed in optical and UV wavelengths with HST and VLT MUSE, ESO 338-4 is ideally suited for multiwavelength feedback studies. We characterize the X-ray emission of ESO 338-4 and its galactic halo using new deep observations obtained with the Chandra X-ray Observatory ( Chandra ) and XMM-Newton . Methods. We analyzed X-ray spectra, light curves, and images of ESO 338-4 to constrain the nature of its X-ray sources. Additionally, we employed photoionization modeling to assess the significance of X-ray sources to the observed nebular He II λ 4686 Å emission. Results. We identified five ultraluminous X-ray sources (ULXs) and diffuse hot gas surrounding ESO 338-4. Two of the ULXs are spatially associated with stellar clusters. The total galactic X-ray luminosity exceeds 10 41 erg s −1 . The brightest point source, ULX1, is variable on timescales of days and is not associated with a stellar cluster. Last, our modeling demonstrates that X-ray sources significantly affect the ionizing photon budget of the galaxy. Photoionization modeling with ULX1 as the ionizing source predicts a high nebular He II λ 4686 Å line luminosity of approximately 10 39 erg s −1 .