  We present new, deep Chandra X-ray and Giant Metrewave Radio Telescope
610~MHz observations of the spiral-galaxy-rich compact group HCG 16, which we
use to examine nuclear activity, star formation and the high luminosity X-ray
binary populations in the major galaxies. We confirm the presence of obscured
active nuclei in NGC 833 and NGC 835, and identify a previously unrecognized
nuclear source in NGC 838. All three nuclei are variable on timescales of
months to years, and for NGC 833 and NGC 835 this is most likely caused by
changes in accretion rate. The deep Chandra observations allow us to detect for
the first time an Fe-K$\alpha$ emission line in the spectrum of the Seyfert 2
nucleus of NGC 835. We find that NGC 838 and NGC 839 are both
starburst-dominated systems, with only weak nuclear activity, in agreement with
previous optical studies. We estimate the star formation rates in the two
galaxies from their X-ray and radio emission, and compare these results with
estimates from the infra-red and ultra-violet bands to confirm that star
formation in both galaxies is probably declining after galaxy-wide starbursts
were triggered ~400-500 Myr ago. We examine the physical properties of their
galactic superwinds, and find that both have temperatures of ~0.8 keV. We also
examine the X-ray and radio properties of NGC 848, the fifth largest galaxy in
the group, and show that it is dominated by emission from its starburst.
