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We combine results from deep ALMA observations of massive (M* > 1010 M⊙) galaxies at different redshifts to show that the column density of their interstellar medium (ISM) rapidly increases toward early cosmic epochs. Our analysis includes objects from the ASPECS and ALPINE large programs, as well as individual observations of z ∼ 6 quasar hosts. When accounting for non-detections and correcting for selection effects, we find that the median surface density of the ISM of the massive galaxy population evolves as ∼(1 + z)3.3. This means that the ISM column density toward the nucleus of a z > 3 galaxy is typically > 100 times larger than locally, and it may reach values as high as Compton-thick at z ≳ 6. Remarkably, the median ISM column density is on the same order of what is measured from X-ray observations of large active galactic nucleus (AGN) samples already at z ≳ 2. We have developed a simple analytic model for the spatial distribution of ISM clouds within galaxies, and estimate the total covering factor toward active nuclei when obscuration by ISM clouds on the host scale is added to that of parsec-scale circumnuclear material (the so-called torus). The model includes clouds with a distribution of sizes, masses, and surface densities, and also allows for an evolution of the characteristic cloud surface density with redshift, Σc, * ∝ (1 + z)γ. We show that, for γ = 2, such a model successfully reproduces the increase in the obscured AGN fraction with redshift that is commonly observed in deep X-ray surveys, both when different absorption thresholds and AGN luminosities are considered. Our results suggest that 80–90% of supermassive black holes in the early Universe (z > 6 − 8) are hidden to our view, primarily by the ISM in their hosts. We finally discuss the implications of our results and how they can be tested observationally with current and forthcoming facilities (e.g., VLT, E-ELT, ALMA, and JWST) and with next-generation X-ray imaging satellites. By extrapolating the observed X-ray nebulae around local AGN to the environments of supermassive black holes at high redshifts, we find ≲1″ nebulae impose stringent design constraints on the spatial resolution of any future X-ray imaging Great Observatory in the coming decades., We acknowledge financial contribution from the agreement ASI-INAF n. 2017-14-H.O. KI acknowledges support by the Spanish MCINN under grant PID2019-105510GB-C33/AEI/10.13039/501100011033 and “Unit of excellence María de Maeztu 2020–2023” awarded to ICCUB (CEX2019-000918-M). SM acknowledges funding from the INAF “Progetti di Ricerca di Rilevante Interesse Nazionale” (PRIN), Bando 2019 (project: “Piercing through the clouds: a multiwavelength study of obscured accretion in nearby supermassive black holes”).