Slice the radishes very thinly - by weight, you should end up with about 95g. Slice the carrot likewise, and weigh that as well: ideally you'll have 95g of carrots, but a little over or under won't matter.
Mix the sliced vegetables well, and pile them into a jar. I used a 12-oz Ball jar here.
Make the pickling brine: mix white vinegar, water, sugar, and salt. Stir until the sugar and salt are completely dissolved.
Pour the brine over the carrots and radishes in the jar. Submerge the veggies completely.
Close the jar with a lid, and refrigerate. The pickles are edible within 2 hours or so, but the flavor improves with an overnight rest.
Store these pickles in the refrigerator, and plan to use them promptly: they'll keep for about 3 weeks.