  We investigate the 5d transition metal oxide BaOsO$_3$ within a combination
of density functional theory (DFT) and dynamical mean-field theory (DMFT),
using a matrix-product-state impurity solver. BaOsO$_3$ has 4 electrons in the
t$_{2g}$ shell akin to ruthenates but stronger spin-orbit coupling (SOC) and is
thus expected to reveal an interplay of Hund's metal behavior with SOC. We
explore the paramagnetic phase diagram as a function of SOC and Hubbard
interaction strengths, identifying metallic, band (van-Vleck) insulating and
Mott insulating regions. At the physical values of the two couplings we find
that BaOsO$_3$ is still situated inside the metallic region and has a moderate
quasiparticle renormalization $m^*/m \approx 2$; consistent with specific heat
measurements. SOC plays an important role in suppressing electronic
correlations (found in the vanishing SOC case) through the splitting of a
van-Hove singularity (vHs) close to the Fermi energy, but is insufficient to
push the material into an insulating van-Vleck regime. In spite of the strong
effect of SOC, BaOsO$_3$ can be best pictured as a moderately correlated Hund's
metal.
