  Classical novae are among the most frequent transient events in the Milky
Way, and key agents of ongoing nucleosynthesis. Despite their large numbers,
they have never been observed in soft $\gamma$-ray emission. Measurements of
their $\gamma$-ray signatures would provide both, insights on explosion
mechanism as well as nucleosynthesis products. Our goal is to constrain the
ejecta masses of $\mathrm{^7Be}$ and $\mathrm{^{22}Na}$ from classical novae
through their $\gamma$-ray line emissions at 478 and 1275 keV. We extract
posterior distributions on the line fluxes from archival data of the
INTEGRAL/SPI spectrometer telescope. We then use a Bayesian hierarchical model
to link individual objects and diffuse emission and infer ejecta masses from
the whole population of classical novae in the Galaxy. Individual novae are too
dim to be detectable in soft $\gamma$-rays, and the upper bounds on their flux
and ejecta mass uncertainties cover several orders of magnitude. Within the
framework of our hierarchical model, we can, nevertheless, infer tight upper
bounds on the $\mathrm{^{22}Na}$ ejecta masses, given all uncertainties from
individual objects as well as diffuse emission, of $<2.0 \times
10^{-7}\,\mathrm{M_{\odot}}$ (99.85th percentile). In the context of ONe
nucleosynthesis, the $\mathrm{^{22}Na}$ bounds are consistent with theoretical
expectations, and exclude that most ONe novae happen on white dwarfs with
masses around $1.35\,\mathrm{M_{\odot}}$. The upper bounds from
$\mathrm{^{7}Be}$ are uninformative. From the combined ejecta mass estimate of
$\mathrm{^{22}Na}$ and its $\beta^+$-decay, we infer a positron production rate
of $<5.5 \times 10^{42}\,\mathrm{e^+\,s^{-1}}$, which would make at most 10% of
the total annihilation rate in the Milky Way.
