We study a class of Dirac semimetals that feature an eightfold-degenerate double Dirac point. We show that 7 of the 230 space groups can host such Dirac points and argue that they all generically display linear dispersion. We introduce an explicit tight-binding model for space groups 130 and 135. Space group 135 can host an intrinsic double Dirac semimetal with no additional states at the Fermi energy. This defines a symmetry-protected topological critical point, and we show that a uniaxial compressive strain applied in different directions leads to topologically distinct insulating phases. In addition, the double Dirac semimetal can accommodate topological line defects that bind helical modes. Connections are made to theories of strongly interacting filling-enforced semimetals, and potential materials realizations are discussed.