  We present an improved determination of the $\pi\pi$ continuum contribution
to the isovector spectral functions of the nucleon electromagnetic form
factors. Our analysis includes the most up-to-date results for the
$\pi\pi\to\bar N N$ partial waves extracted from Roy-Steiner equations,
consistent input for the pion vector form factor, and a thorough discussion of
isospin-violating effects and uncertainty estimates. As an application, we
consider the $\pi\pi$ contribution to the isovector electric and magnetic radii
by means of sum rules, which, in combination with the accurately known neutron
electric radius, are found to slightly prefer a small proton charge radius.
