Quasi-periodic eruptions (QPEs) are high-amplitude, soft X-ray bursts recurring every few hours, associated with supermassive black holes. Many interpretations for QPEs were proposed since their recent discovery in 2019, including extreme mass ratio inspirals and accretion disk instabilities. But, as of today, their nature still remains debated. We performed the first high-resolution X-ray spectral study of a QPE source using the RGS gratings onboard XMM-Newton, leveraging in total nearly 2 Ms of exposure on GSN 069, the first discovered source of this class. We resolved several absorption and emission lines including a strong line pair near the N VII rest-frame energy, resembling the P-Cygni profile. We applied photoionization spectral models and identified the absorption lines as an outflow blueshifted by about 2500 km/s. The emission lines are instead redshifted by up to 3000 km/s, and likely originate from the same outflow that imprints the absorption features, and covers the full 4π sky from the point of view of GSN 069. From the properties of the outflow we derived limits on its location which indicate that the outflow is connected to the recent complex, transient activity of GSN 069 that began around 2010.