Trim carrot tops, if using, to about 1/2 in., and trim root hairs, then cut carrots in half lengthwise. Cut fennel lengthwise into slender wedges. Set vegetables in a large bowl of ice water about 1 hour to crisp.
Meanwhile, if using baby turnips, trim tops to about 1/2 in., then quarter turnips; or cut regular turnips into slender wedges. Remove all but a few small leaves on each radish; swish radishes in water to clean. Crisp turnips and radishes in another bowl of ice water about 45 minutes.
Drain vegetables and arrange on a large platter. (Drape with damp towels, if they'll be standing before serving.) Serve with oil for dunking.