We present the analysis of Chandra and NuSTAR spectra of NGC 4968, a local (D$sim$44 Mpc) 12$mu$m-selected Seyfert 2 galaxy, enshrouded within Compton-thick layers of obscuring gas. We find no evidence of variability between the Chandra and NuSTAR observations (separated by 2 years), and between the two NuSTAR observations (separated by 10 months). Using self-consistent X-ray models, we rule out the scenario where the obscuring medium is nearly spherical and uniform, contradicting the results implied by the $<$10 keV Chandra spectrum. The line-of-sight column density, from intervening matter between the source and observer that intercepts the intrinsic AGN X-ray emission, is well within the Compton-thick regime, with a minimum column density of $2times10^{24}$ cm$^{-2}$. The average global column density is high ($> 3times10^{23}$ cm$^{-2}$), with both Compton-thick and Compton-thin solutions permitted depending on the X-ray spectral model. The spectral models provide a range of intrinsic AGN continuum parameters and implied 2-10 keV luminosities ($L_{rm 2-10keV,intrinsic}$), where the higher end of $L_{rm 2-10keV,intrinsic}$ is consistent with expectations from the 12$mu$m luminosity ($L_{rm 2-10keV,intrinisc} sim 7times10^{42}$ erg s$^{-1}$). Compared with Compton-thick AGN previously observed by {it NuSTAR}, NGC 4968 is among the most intrinsically X-ray luminous. However, despite its close proximity and relatively high intrinsic X-ray luminosity, it is undetected by the 105 month Swift-BAT survey, underscoring the importance of multi-wavelength selection for obtaining the most complete census of the most hidden black holes.
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