We report ${}^{23}$Na-NMR study of the alkali superoxide NaO${}_2$, a candidate one-dimensional quantum spin system. The Knight shift, linewidth, and spin-lattice relaxation rate $1/T_1$ were investigated down to 0.3 K in magnetic fields up to 16 T. Below $T_{\rm S3} =$ 34 K, $1/T_1$ exhibits thermally activated behavior, indicating the opening of a spin gap of $\Delta(10.1 {\rm T}) \simeq$ 38.0 K and $\Delta(3.55 {\rm T}) \simeq$ 46 K. The hyperfine coupling constant shows a sudden reduction across the structural transition at $T_{\rm S2} \simeq$ 195 K, reflecting the development the pronounced one-dimensionality in the elctron structure. These results provide microscopic evidence that NaO${}_2$ is a rare $\pi$-orbital-based one-dimensional spin chain that exhibits a spin-gapped ground state, consistent with a spin-Peierls-like instability.