Actually depleted uranium is still radioactive "The enrichment process reduces the radioactivity of depleted uranium to approximately half of that of natural uranium".
Apart from ammunition it is used for shielding isotopes because it is denser than lead or tungsten, obviously higher density is better for ammo also. In my job it is used to shield Iridium and Selenium sources, with the sources out the DU container still emits.
You have basically 2 types of radiation to be concerned about, radiation exposure and radiation contamination.
Exposure is the electromagnetic waves passing through your body like from an xray machine or isotope. Isotopes are split into Gamma, Beta, Neutron and Alpha. For exposure the larger ones are Gamma and Neutron as the pass through you at the speed of light and being large can juggle long chain molecules about i.e. DNA, sperm, white blood cells etc. The intensify of the exposure is 4 times higher if you halve the distance between you and it, so if it is in you hand the calculation gets interestingly high because even low intensities get rather high. Anyway lots of extra calculations are possible like whole body exposure or just hand etc.
The biggest problem with depleted uranium is contamination as we are talking about digestion or internal contamination. DU on impact creates dust and it also corrodes. If this dust is ingested the oxides emit Alpha particles. In normal air a piece of paper stops them but if ingested they create the most damage to cell structure. It is quite a complex problem but the paper attached discusses the magnitude of DU contamination.
toxics-02-00050.pdf