LiZn$_2$Mo$_3$O$_8$ has been proposed to contain $S~=~1/2$ Mo$_3$O$_{13}$ magnetic clusters arranged on a triangular lattice with antiferromagnetic nearest-neighbor interactions. Here, microwave and terahertz electron spin resonance (ESR), $^7$Li nuclear magnetic resonance (NMR), and muon spin rotation ($mu textrm{SR}$) spectroscopies are used to characterize the local magnetic properties of LiZn$_2$Mo$_3$O$_8$. These results show the magnetism in LiZn$_2$Mo$_3$O$_8$ arises from a single isotropic $S~=~1/2$ electron per cluster and that there is no static long-range magnetic ordering down to $T~=~0.07,textrm{K}$. Further, there is evidence of gapless spin excitations with spin fluctuations slowing down as the temperature is lowered. These data indicate strong spin correlations which, together with previous data, suggest a low-temperature resonating valence-bond state in LiZn$_2$Mo$_3$O$_8$.
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