NGC 4945 is one of the brightest Seyfert galaxies on the sky at 100 keV, but is completely absorbed below 10 keV; its absorption column is probably the largest that still allows a direct view of the nucleus at hard X-ray energies. Our observations of it with the Rossi X-Ray Timing Explorer (RXTE) satellite confirm the large absorption, which for a simple phenomenological fit using an absorber with solar abundances implies a column of 4.5+0.4-0.4x1024 cm(-2). Using a more realistic scenario (requiring Monte Carlo modeling of the scattering), we infer the optical depth to Thomson scattering of approximately 2.4. If such a scattering medium were to subtend a large solid angle from the nucleus, it should smear out any intrinsic hard X-ray variability on timescales shorter than the light-travel time through it. The rapid (with a timescale of approximately 1 day) hard X-ray variability of NGC 4945 discovered by us with RXTE implies that the bulk of the extreme absorption in this object does not originate in a parsec-size, geometrically thick molecular torus. Instead, the optically thick material on parsec scales must be rather geometrically thin, subtending a half-angle less than 10 degrees, and it is likely to be the same disk of material that is responsible for the water maser emission observed in NGC 4945. Local number counts of Seyfert 1 and Seyfert 2 galaxies show a large population of heavily obscured active galactic nuclei (AGNs) which are proposed to make up the cosmic X-ray background (CXRB). However, for this to be the case, the absorption geometry in the context of axially symmetric unification schemes must have the obscuring material subtending a large scale height-contrary to our inferences about NGC 4945-implying that NGC 4945 is not a prototype of obscured AGNs postulated to make up the CXRB. The small solid angle of the absorber, together with the black hole mass (of approximately 1.4x106 M( middle dot in circle)) from megamaser measurements, allows a robust determination of the nuclear luminosity, which in turn implies that the source radiates at approximately 10% of the Eddington limit.