The distribution of neutrons with energies below 15 MeV in spherical stony meteoroids is calculated using the ANISN neutron-transport code. The source distributions and intensities of neutrons are calculated using cross sections for the production of tritium. The meteoroid's radius and chemical composition strongly influence the total neutron flux and the neutron energy spectrum, whereas the location within a meteoroid only affects the relative neutron intensities. Meteoroids must have radii of more than 50 g/cm2 before they have appreciable fluxes of neutrons near thermal energies. Meteoroids with high hydrogen or low iron contents can thermalize neutrons better than chondrites can. Rates for the production of 60Co, 59Ni, and 36Cl are calculated with these neutron fluxes and evaluated neutron-capture cross sections and are reported for carbonaceous chondrites with high hydrogen contents, L-chondrites, and aubrites. For most meteoroids with radii <300 g/cm2, the production rates of these neutron-capture nuclides increase monotonically with depth. The highest calculated 60Co production rate in an ordinary chondrite is 375 atoms/min/g-Co at the center of a meteoroid with a 250 g/cm2 radius. The production rates calculated for spallogenic 60Co and 59Ni are greater than the neutron-capture rates for radii less than ~50-75 g/cm2. Only for very large meteoroids and chlorine-rich samples is the neutron-capture production of 36Cl important. The results of these calculations are compared with those of previous calculations and with measured activities in many meteorites.