We report the existence of confined massless fermion states in a graphene quantum well (QW) by means of analytical and numerical calculations. These states show an unusual quasilinear dependence on the momentum parallel to the QW: their number depends on the wave vector and is constrained by electron-hole conversion in the barrier regions. An essential difference with nonrelativistic electron states is a mixing between free and confined states at the edges of the free-particle continua, demonstrated by the direction-dependent resonant transmission across a potential well.