  Measurement of the largest angular scale ($\ell < 30$) features of the cosmic
microwave background (CMB) polarization is a powerful way to constrain the
optical depth to reionization and search for the signature of inflation through
the detection of primordial $B$-modes. We present an analysis of maps covering
73.6\% of the sky made from the $40\,\mathrm{GHz}$ channel of the Cosmology
Large Angular Scale Surveyor (CLASS) from 2016 August to 2022 May. Taking
advantage of the measurement stability enabled by front-end polarization
modulation and excellent conditions from the Atacama Desert, we show this
channel achieves higher sensitivity than the analogous frequencies from
satellite measurements in the range $10 < \ell < 100$. Simulations show the
CLASS linear (circular) polarization maps have a white noise level of $125
\,(130)\,\mathrm{\mu K\, arcmin}$. We measure the Galaxy-masked $EE$ and $BB$
spectra of diffuse synchrotron radiation and compare to space-based
measurements at similar frequencies. In combination with external data, we
expand measurements of the spatial variations of the synchrotron spectral
energy density (SED) to include new sky regions and measure the diffuse SED in
the harmonic domain. We place a new upper limit on a background of circular
polarization in the range $5 < \ell < 125$ with the first bin showing $D_\ell <
0.023$ $\mathrm{\mu K^2_{CMB}}$ at 95\% confidence. These results establish a
new standard for recovery of the largest-scale CMB polarization from the ground
and signal exciting possibilities when the higher sensitivity and
higher-frequency CLASS channels are included in the analysis.
