  There is a long-standing discrepancy between different measurements of the
electric and magnetic radii of the proton. Lattice QCD calculations are a
well-suited tool for theoretical investigations of the structure of the nucleon
from first principles. However, all previous lattice studies of the proton's
electromagnetic radii have either neglected quark-disconnected contributions or
were not extrapolated to the continuum and infinite-volume limit. Here, we
present results for the electromagnetic form factors of the proton and neutron
computed on the $(2 + 1)$-flavor Coordinated Lattice Simulations (CLS)
ensembles including both quark-connected and -disconnected contributions. From
simultaneous fits to the $Q^2$-, pion-mass, lattice-spacing, and finite-volume
dependence of the form factors, we determine the electric and magnetic radii
and the magnetic moments of the proton and neutron. For the proton, we obtain
as our final values $\langle r_E^2 \rangle^p = (0.672 \pm 0.014$ (stat)${} \pm
0.018$ (syst)$)$ fm${}^2$, $\langle r_M^2 \rangle^p = (0.658 \pm 0.012$
(stat)${} \pm 0.008$ (syst)$)$ fm${}^2$, and $\mu_M^p = 2.739 \pm 0.063$
(stat)${} \pm 0.018$ (syst). The magnetic moment is in good agreement with the
experimental value, as is the one of the neutron. On the one hand, our result
for the electric (charge) radius of the proton clearly points towards a small
value, as favored by muonic hydrogen spectroscopy and the recent
$ep$-scattering experiment by PRad. Our estimate for the magnetic radius, on
the other hand, is well compatible with that inferred from the A1
$ep$-scattering experiment.
