If you're like my husband, keeping radishes fresh for any length of time is not an issue. If you are like him, you will eat them straight out of the CSA box (maybe you will wash them), you will dip them in a little salt and polish off the whole bunch while you stand over the kitchen sink. However, if you're not like my husband, here's a little how-to on keeping radishes fresh in your refrigerator.
Now, for the first few days you can just store them, unwashed, in your crisper drawer. Once the tops start to look a little wilted, that's your cue to take those suckers, cut off the root and stem ends of each, and throw them in a jar. Fill the jar with enough water to cover the radishes and stick the jar back in the fridge. This will keep them fresh for at least a week, if not longer. An old salsa jar or mason jar both work great for this, but really, even a large drinking glass will work, depending on how many radishes you have.
(Of course, I never need to use this method, because radishes disappear from my fridge in about three seconds flat--if they ever make it in there at all.)