We present absorption-line index gradients for a sample of S0 galaxies in the Fornax cluster. The sample has been selected to span a wide range in galaxy mass, and the deep FORS2 spectroscopy on the VLT allows us to explore the stellar populations all the way to the outer disc-dominated regions of these galaxies. We find that globally, in both bulges and discs, star formation ceased earliest in the most massive systems, as a further manifestation of downsizing. However, within many galaxies, we find an age gradient which indicates that star formation ended first in the outermost regions. Metallicity gradients, when detected, are always negative such that the galaxy centres are more metal-rich. This finding fits with a picture in which star formation continued in the central regions, with enriched material, after it had stopped in the outskirts. Age and metallicity gradients are correlated, suggesting that large differences in star formation history between the inner and outer parts of S0 galaxies yield large differences in their chemical enrichment. In agreement with previous results, we conclude that the radial variations in the stellar populations of S0 galaxies are compatible with the hypothesis that these galaxies are the descendants of spiral galaxies whose star formation has ceased. With the addition of radial gradient information, we are able to show that this shutdown of star formation occurred from the outside inward, with the later star formation in the central regions offering a plausible mechanism for enhancing the bulge light in these systems, as the transformation to more bulge-dominated S0 galaxies requires.