  The origins of quasi-periodic eruptions (QPEs) are poorly understood,
although most theoretical explanations invoke an accretion disk around a
supermassive black hole. The gas and stellar environments in the galactic
nuclei of these sources are also poorly constrained. In this paper, we present
an analysis of archival Hubble Space Telescope (HST) images to study the
narrow-line [O III] emission in the QPE source GSN 069. We find strong evidence
for a compact nuclear [O III] emission region of size $\lesssim 35$ pc,
overlaid on top of extended [O III] emission up to 2 kpc away from the nucleus.
The age of the accretion system is estimated to be between 10 and 100 yr. The
[O III] luminosity of the compact region was measured to be $(2.1 \pm 0.3)
\times 10^{40}\,\rm erg\,s^{-1}$. Based on CLOUDY simulations, we constrain
that the [O III] emitting gas has a hydrogen number density in the range $5
\times 10^{3} < n_{\rm H} \lesssim 10^{8}\,\rm cm^{-3}$ and volume filling
factor $f_{\rm V} < 2 \times 10^{-3}$. We suggest that the dense gas in the
nuclear region of GSN 069 originates from molecular clouds (with total mass
$\gtrsim 3 \times 10^{3}\,M_{\odot}$), which are freshly ionised by the soft
X-ray photons from the accretion disk. We predict possible evolution of the
compact narrow-line region on emission-line diagnostic diagrams, and hence
future HST or integral-field unit observations can be used to further pin down
the age of this puzzling system.
