Carrots and subterranean vegetables can be grow hydroponically as well, hydro is not just limited to above soil growth.
What you will need:
Vermiculite - add to perlite for better water retention, not absolutely necessary but it does help.
A good deep, ideally lightproof plastic container with holes at the bottom - at least 12 inches deep and the same wide. The bigger the better.
Another plastic container that the one above can sit in, this will hold a quantity of water all the time. Alternatively, a deep drip tray - say 2 to 3 inches in depth.
Fill your container with perlite, ideally you would add vermiculite to this as well - the ratio is 3:1 perlite to vermiculite.
Place your seeds in the main container, cover with vermiculite to the depth that is written on your seed packet.
Lightly mist/spray the top of your main container with some water, we need to get some water to the seeds to start. Keep the top of the main container moist, keep misting as appropriate until the seeds establish themselves - see your seed packet for germination times.
Don't pour water directly down on them at this phase, the seeds will be washed away by this!
Once the seeds have germinated, give the top of this container a good watering three times a day - approx.
Keep the drip tray (or other container), full of water
Generally - carrots take 2-3 months to mature and germinate in 6 to 10 days.
Radishes are also a good crop, they need less space and grow really quickly. They will mature in 30 to 40 days and germinate in 2 to 6 days.
Enjoy the world of subterranean hydo...
Edit: Gisette edited the thumbnail links for better website performance (they were grabbing the full-size pictures and scaling them into thumbnails - not too quick).