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 \sim 400-500 Myr ago . We examine the physical properties of their galactic superwinds , and find that both have temperatures of \sim 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 .