Spin-orbit coupling (SOC) often gives rise to interesting electronic and magnetic phases in an otherwise ordinary pool of paramagnetic heavy metal oxides. In presence of strong SOC, assumed to be working in $j$-$j$ coupling regime, 5$d^4$ iridates are generally speculated to possess a nonmagnetic $J_{eff}$~=~0 singlet ground state, which invariably gets masked due to different solid-state effects (e.g. hopping). Here, we try to probe the trueness of the atomic SOC-based proposal in an apparently 1-dimensional system, Sr$_3$NaIrO$_6$, possessing a 2$H$ hexagonal structure with well separated Ir$^{5+}$ (5$d^4$) ions. But all the detailed experimental as well as theoretical characterizations reveal that the ground state of Sr$_3$NaIrO$_6$ is not nonmagnetic, rather accommodating a significantly high effective magnetic moment on Ir$^{5+}$ ion. However our combined dc susceptibility ($chi$), ${}^{23}$Na nuclear magnetic resonance (NMR), muon-spin-relaxation/rotation ($mu$SR) and heat capacity ($C_p$) measurements clearly refute any sign of spin-freezing or ordered magnetism among the Ir$^{5+}$ moments due to geometrical exchange frustration, while in-depth zero-field (ZF) and longitudinal field (LF) $mu$SR investigations strongly point towards inhomogeneous quantum spin-orbital liquid (QSOL)-like ground state. In addition, the linear temperature dependence of both the NMR spin-lattice relaxation rate and the magnetic heat capacity at low temperatures suggest low-lying gapless spin excitations in the QSOL phase of this material. Finally, we conclude that the effective SOC realised in $d^4$ iridates are unlikely to offer a ground state which will be consistent with a purely atomic $j$-$j$ coupling description.
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