I have not done root vegetables hydroponically, but I don't think I would use a flood and drain type of system for them. The roots system would be submerged in water to long to be good for the vegetables I would think. I would use a
drip system. I attached a quick drawing of a box to give you an idea. Just find a container the size you need and run a line back to the reservoir at the bottom. Run drip irrigation along the top of the
growing medium. This way the roots wont be swimming in water and would just trickle down the same way it does in mother nature.
I would probably use coco fiber and or coco chips for a growing medium because it holds moisture well and I am familiar with it, though Perlite and Vermiculite would probably do well also. You just want to make sure that the growing medium does not make it down the drain spout. you could line the bottom of the container with a few layers of cheesecloth or screen door screening as a mesh to strain it and keep the growing medium in place.