We demonstrate a compact magneto-optical trap (MOT) of alkaline-earth atoms using a nanofabricated diffraction grating chip. A single input laser beam, resonant with the broad S-1(0) -> P-1(1) transition of strontium, forms the MOT in combination with three diffracted beams from the grating chip and a magnetic field produced by permanent magnets. A differential pumping tube limits the effect of the heated, effusive source on the background pressure in the trapping region. The system has a total volume of around 2.4 l. With our setup, we have trapped up to 5 x 10(6) Sr-88 atoms at a temperature of similar to 6 mK, and with a trap lifetime of similar to 1 s. Our results will aid the effort to miniaturize quantum technologies based on alkaline-earth atoms.