  Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) offers a unique way
to study neutrino properties and to search for new physics beyond the Standard
Model. Nuclear reactors are promising sources to explore this process at low
energies since they deliver large fluxes of (anti-)neutrinos with typical
energies of a few MeV. In this paper, a new-generation experiment to study
CE$\nu$NS is described. The NUCLEUS experiment will use cryogenic detectors
which feature an unprecedentedly low energy threshold and a time response fast
enough to be operated in above-ground conditions. Both sensitivity to
low-energy nuclear recoils and a high event rate tolerance are stringent
requirements to measure CE$\nu$NS of reactor antineutrinos. A new experimental
site, denoted the Very-Near-Site (VNS) at the Chooz nuclear power plant in
France is described. The VNS is located between the two 4.25 GW$_{\mathrm{th}}$
reactor cores and matches the requirements of NUCLEUS. First results of on-site
measurements of neutron and muon backgrounds, the expected dominant background
contributions, are given. In this paper a preliminary experimental setup with
dedicated active and passive background reduction techniques is presented.
Furthermore, the feasibility to operate the NUCLEUS detectors in coincidence
with an active muon-veto at shallow overburden is studied. The paper concludes
with a sensitivity study pointing out the promising physics potential of
NUCLEUS at the Chooz nuclear power plant.
