We present a multiwavelength study of seven active galactic nuclei (AGN) at spectroscopic redshift >2.5 in the 7 Ms Chandra Deep Field South that were selected for their good far-infrared (FIR) and submillimeter (submm) detections. Our aim is to investigate the possibility that the obscuration observed in the X-rays can be produced by the interstellar medium (ISM) of the host galaxy. Based on the 7 Ms Chandra spectra, we measured obscuring column densities N_{H, X} in excess of 7 × 10²² cm^-2 and intrinsic X-ray luminosities L_X > 10⁴⁴ erg s^-1 for our targets, as well as equivalent widths for the Fe Kα emission line EW_rest ≳ 0.5-1 keV. We built the UV-to-FIR spectral energy distributions (SEDs) by using broadband photometry from the CANDELS and Herschel catalogs. By means of an SED decomposition technique, we derived stellar masses (M_* ∼ 10¹¹ M_⊙), IR luminosities (L_IR > 10¹² L_⊙), star formation rates (SFR ∼ 190-1680 M_⊙ yr^-1) and AGN bolometric luminosities (L_bol ∼ 10⁴⁶ erg s^-1) for our sample. We used an empirically calibrated relation between gas masses and FIR/submm luminosities and derived M_gas ∼ 0.8-5.4 × 10¹⁰ M_⊙. High-resolution (0.3-0.7″) ALMA data (when available, CANDELS data otherwise) were used to estimate the galaxy size and hence the volume enclosing most of the ISM under simple geometrical assumptions. These measurements were then combined to derive the column density associated with the ISM of the host, which is on the order of N_{H, ISM} ∼ 10^23-24 cm^-2. The comparison between the ISM column densities and those measured from the X-ray spectral analysis shows that they are similar. This suggests that at least at high redshift, significant absorption on kiloparsec scales by the dense ISM in the host likely adds to or substitutes that produced by circumnuclear gas on parsec scales (i.e., the torus of unified models). The lack of unobscured AGN among our ISM-rich targets supports this scenario.