  We determine the radii and masses of 293 nearby, bright M dwarfs of the
CARMENES survey. This is the first time that such a large and homogeneous
high-resolution (R>80 000) spectroscopic survey has been used to derive these
fundamental stellar parameters. We derived the radii using Stefan-Boltzmann's
law. We obtained the required effective temperatures $T_{\rm eff}$ from a
spectral analysis and we obtained the required luminosities L from integrated
broadband photometry together with the Gaia DR2 parallaxes. The mass was then
determined using a mass-radius relation that we derived from eclipsing binaries
known in the literature. We compared this method with three other methods: (1)
We calculated the mass from the radius and the surface gravity log g, which was
obtained from the same spectral analysis as $T_{\rm eff}$. (2) We used a widely
used infrared mass-magnitude relation. (3) We used a Bayesian approach to infer
stellar parameters from the comparison of the absolute magnitudes and colors of
our targets with evolutionary models. Between spectral types M0V and M7V our
radii cover the range $0.1\,R_{\normalsize\odot}<R<0.6\,R_{\normalsize\odot}$
with an error of 2-3% and our masses cover $0.09\,{\mathcal
M}_{\normalsize\odot}<{\mathcal M}<0.6\,{\mathcal M}_{\normalsize\odot}$ with
an error of 3-5%. We find good agreement between the masses determined with
these different methods for most of our targets. Only the masses of very young
objects show discrepancies. This can be well explained with the assumptions
that we used for our methods.
