  [Abridged] We exploit the extreme radial velocity (RV) precision of the
ultra-stable echelle spectrograph ESPRESSO on the VLT to unveil the physical
properties of the transiting sub-Neptune TOI-130 b, uncovered by TESS orbiting
the nearby, bright, late F-type star HD 5278 (TOI-130) with a period $P_{\rm
b}=14.3$. We use 43 ESPRESSO high-resolution spectra and broad-band photometry
information to derive accurate stellar atmospheric and physical parameters of
HD 5278. We exploit the TESS light curve (LC) and spectroscopic diagnostics to
gauge the impact of stellar activity on the ESPRESSO RVs. We perform a joint
ESPRESSO RVs + TESS LC analysis using fully Bayesian frameworks to determine
the system parameters. The updated stellar parameters of HD 5278 are
T$_\mathrm{eff}=6203\pm64$ K, $\log g =4.50\pm0.11$ dex, [Fe/H]=$-0.12\pm0.04$
dex, M$_\star=1.126_{-0.035}^{+0.036}$ M$_\odot$ and
R$_\star=1.194_{-0.016}^{+0.017}$ R$_\odot$. We determine HD 5278 b's mass and
radius to be $M_{\rm b} = 7.8_{-1.4}^{+1.5}$ M$_\oplus$ and $R_{\rm b} =
2.45\pm0.05$ R$_\oplus$. The derived mean density, $\varrho_{\rm b} =
2.9_{-0.5}^{+0.6}$ g cm$^{-3}$, is consistent with a bulk composition with a
substantial ($\sim30\%$) water mass fraction and a gas envelope comprising
$\sim17\%$ of the measured radius. Given the host brightness and irradiation
levels, HD 5278 b is one of the best targets orbiting G-F primaries for
follow-up atmospheric characterization measurements with HST and JWST. We
discover a second, non-transiting companion in the system, with a period
$P_{\rm c}=40.87_{-0.17}^{+0.18}$ days and a minimum mass $M_{\rm c}\sin i_{\rm
c} =18.4_{-1.9}^{+1.8}$ M$_\oplus$. We study emerging trends in the growing
population of transiting sub-Neptunes, and provide statistical evidence for a
low occurrence of close-in, $10-15$ M$_\oplus$ companions around G-F primaries
with $T_\mathrm{eff}\gtrsim5500$ K.
