  By means of density functional theory plus dynamical mean-field theory
(DFT+DMFT) calculations and resonant inelastic x-ray scattering (RIXS)
experiments, we investigate the high-pressure phases of the spin-orbit-coupled
$J_{\rm{eff}}=3/2$ insulator GaTa$_4$Se$_8$. Its metallic phase, derived from
the Mott state by applying pressure, is found to carry $J_{\rm{eff}}=3/2$
moments. The characteristic excitation peak in the RIXS spectrum maintains its
destructive quantum interference of $J_{\rm{eff}}$ at the Ta $L_2$-edge up to
10.4 GPa. Our exact diagonalization based DFT+DMFT calculations including
spin-orbit coupling also reveal that the $J_{\rm{eff}}=3/2$ character can be
clearly identified under high pressure. These results establish the intriguing
nature of the correlated metallic magnetic phase, which represents the first
confirmed example of $J_{\rm{eff}}$=3/2 moments residing in a metal. They also
indicate that the pressure-induced superconductivity is likely unconventional
and influenced by these $J_{\rm{eff}}=3/2$ moments. Based on a self-energy
analysis, we furthermore propose the possibility of doping-induced
superconductivity related to a spin-freezing crossover.
