Correlated electron systems are highly susceptible to various forms of electronic order. By tuning the transition temperature towards absolute zero, striking deviations from conventional metallic (Fermi-liquid) behaviour can be realized. Evidence for electronic nematicity, a correlated electronic state with broken rotational symmetry, has been reported in a host of metallic systems1-5 that exhibit this so-called quantum critical behaviour. In all cases, however, the nematicity is found to be intertwined with other forms of order, such as antiferromagnetism5-7 or charge-density-wave order8, that might themselves be responsible for the observed behaviour. The iron chalcogenide FeSe1-xSx is unique in this respect because its nematic order appears to exist in isolation9-11, although until now, the impact of nematicity on the electronic ground state has been obscured by superconductivity. Here we use high magnetic fields to destroy the superconducting state in FeSe1-xSx and follow the evolution of the electrical resistivity across the nematic quantum critical point. Classic signatures of quantum criticality are revealed: an enhancement in the coefficient of the T2 resistivity (due to electron-electron scattering) on approaching the critical point and, at the critical point itself, a strictly T-linear resistivity that extends over a decade in temperature T. In addition to revealing the phenomenon of nematic quantum criticality, the observation of T-linear resistivity at a nematic critical point also raises the question of whether strong nematic fluctuations play a part in the transport properties of other 'strange metals', in which T-linear resistivity is observed over an extended regime in their respective phase diagrams.