Aims: 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. Methods: We derived the radii using Stefan-Boltzmann's law. We obtained the required effective temperatures Teff 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 Teff. (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. Results: Between spectral types M0 V and M7 V our radii cover the range 0.1 R⊙ < R < 0.6 R⊙ with an error of 2-3% and our masses cover 0.09 ℳ⊙ <ℳ< 0.6ℳ⊙ 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. Table B.1 (stellar parameters) is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/625/A68