We present a dispersion theoretical analysis of the experimental data on the electromagnetic form factors of the nucleon covering both the space- and time-like regions. The nucleon form factors over the full range of momentum transfers and the nucleon radius are extracted with high precision. The statistical uncertainties of the extracted form factors and radius are estimated using the bootstrap method, while systematic errors are determined from variations of the spectral functions. For the proton charge radius, we find $r_E^p = 0.840^{+0.003}_{-0.002}{}^{+0.002}_{-0.002}$~fm, in line with previous analyses of spacelike data alone and also the muonic Lamb shift determination.