  We construct two-band topological semimetals in four dimensions using the
unstable homotopy of maps from the three-torus $T^3$ (Brillouin zone of a 3D
crystal) to the two-sphere $S^2$. Dubbed ``Hopf semimetals'', these gapless
phases generically host nodal lines, with a surface enclosing such a nodal line
in the four-dimensional Brillouin zone carrying a Hopf flux. These semimetals
show a unique class of surface states: while some three-dimensional surfaces
host gapless Fermi-arc states {\em and} drumhead states, other surfaces have
gapless Fermi surfaces. Gapless two-dimensional corner states are also present
at the intersection of three-dimensional surfaces.
