In this paper, we study the nuclear obscuration of galaxies hosting low ionization narrow emission regions (LINERs) based on their X-ray and optical emission. They show column densities at soft energies (0.5-2 keV) mostly related to the diffuse emission around the active galactic nucleus (AGN), showing a correlation with the optical extinction. Column densities at hard energies (2-10 keV) seem to be much higher than what would be expected from the optical extinction. They might be associated with the inner regions of the AGN, buried at optical wavelengths. The main result of this paper is that around 50% of our LINER sample shows signatures of Compton-thickness according to the most common tracers: the X-ray spectral index, F_{X}(2-10 keV)/F([O III]) ratio, and FeKα equivalent width (EW). However, the EWs of the Compton-thick LINERs are significantly lower than in the Compton-thick Seyferts (sime200 eV against >=500 eV), suggesting that the 2-10 keV emission is dominated by electron scattering of the otherwise invisible AGN, or by emission from shocked gas associated with star formation rather than by reflection from the inner wall of the torus. However, no clear relation seems to exist between galaxies with optical dust lanes and X-ray classified Compton-thick objects. This may suggest that Compton-thick sources should be related to absorbing material located at the very inner regions of the AGN, maybe in the putative dusty torus. Larger black hole masses and lower Eddington ratios than Seyfert galaxies have been found. This effect can be better attributed to LINER nuclei being hosted by earlier morphological types than Seyfert nuclei. However, it has to be noted that, once a proper correction to the X-ray luminosity is applied, LINERs show Eddington ratios overlapping those of type 2 Seyferts. We speculate with a possible scenario for LINER nuclei: an inner obscuring matter similar to that of type 2 Seyfert, and an external obscuring matter responsible for the optical extinction. Compton-thick sources appear to be more common among LINERs than Seyferts.