We show that the radial profiles of the temperature and density of the electrons as well as the magnetic field strength around the massive black hole at the Galactic center, Sgr A*, may be constrained directly from existing radio data without any need to make prior assumptions about the dynamics of the emitting gas. The observed spectrum and wavelength-dependent angular size of Sgr A* indicate that the synchrotron emission originates from an optically-thick plasma of quasi-thermal electrons. We find that the electron temperature rises above the virial temperature within tens of Schwarzschild radii from the black hole, suggesting that the emitting plasma may be outflowing. Constraints on the electron density profile are derived from polarization measurements. Our best-fit results differ from expectations based on existing theoretical models. However, these models cannot be ruled out as of yet due to uncertainties in the source size measurements. Our constraints could tighten considerably with future improvements in the size determination and simultaneous polarization measurements at multiple wavelengths.