  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 perform the first
high-resolution X-ray spectral study of a QPE source using the RGS gratings
onboard XMM-Newton, leveraging nearly 2 Ms of exposure on GSN 069, the first
discovered source of this class. We resolve several absorption and emission
lines including a strong line pair near the N VII rest-frame energy, resembling
the P-Cygni profile. We apply photoionization spectral models and identify the
absorption lines as an outflow blueshifted by $1700-2900$ km/s, with a column
density of about $10^{22}$ cm$^{-2}$ and an ionization parameter $\log
(\xi$/erg cm s$^{-1})$ of $3.9-4.6$. The emission lines are instead redshifted
by up to 2900 km/s, and likely originate from the same outflow that imprints
the absorption features, and covers the full $4\pi$ sky from the point of view
of GSN 069. The column density and ionization are comparable to the outflows
detected in some tidal disruption events, but this outflow is significantly
faster and has a strong emission component. The outflow is more highly ionized
when the system is in the phase during which QPEs are present, and from the
limits we derive on its location, we conclude that the outflow is connected to
the recent complex, transient activity of GSN 069 which began around 2010.
