Plasmonic response of the metallic structure characterized by sub-nanometer dielectric gaps can be strongly affected by nonlocal or quantum effects. In this paper, we investigate these effects in spherical Na and Au nanomatryoshka structures with sub-nanometer core-shell separation. We use the state-of-the-art quantum hydrodynamic theory (QHT) to study both near-field and far-field optical properties of these systems: results are compared with the classical local response approximation (LRA), Thomas-Fermi hydrodynamic theory (TF-HT), and the reference time-dependent density functional theory (TD-DFT). We find that the results obtained using the QHT method are in a very good agreement with TD-DFT calculations, whereas other LRA and TF-HT significantly overestimate the field-enhancements. Thus, the QHT approach efficiently and accurately describes microscopic details of multiscale plasmonic systems whose sizes are computationally out-of-reach for a TD-DFT approach; here, we report results for Na and Au nanomatryoshka with a diameter of 60 nm.