We consider the dark matter (DM) scenario in the context of the classically conformal U(1)' extended standard model (SM), with three right-handed neutrinos (RHNs) and the U(1)' Higgs field. The model is free from all of the U(1)' gauge and gravitational anomalies in the presence of the three RHNs. We introduce a Z(2) parity in the model, under which an odd parity is assigned to one RHN, while all of the other particles are assigned to be Z(2) even, and hence the Z(2)-odd RHN serves as a DM candidate. In this model, the U(1)' gauge symmetry is radiatively broken through the Coleman-Weinberg mechanism, by which the electroweak symmetry breaking is triggered. There are three free parameters in our model-the U(1)' charge of the SM Higgs doublet (x(H)), the new U(1)' gauge coupling (g(X)), and the U(1)' gauge boson (Z') mass (m(Z'))-which are severely constrained in order to solve the electroweak vacuum instability problem, and satisfy the LHC Run-2 bounds from the search for the Z' boson resonance. In addition to these constraints, we investigate the RHN DM physics. Because of the nature of classical conformality, we find that a RHN DM pair mainly annihilates into the SM particles through Z' boson exchange. This is the so-called Z'-portal DM scenario. Combining the electroweak vacuum stability condition, the LHC Run-2 bounds, and the cosmological constraint from the observed DM relic density, we find that all constraints work together to narrow the allowed parameter regions and, in particular, exclude m(Z') less than or similar to 3.5 TeV. For the obtained allowed regions, we calculate the spin-independent cross section of the RHN DM with nucleons. We find that the resultant cross section is well below the current experimental upper bounds.