Many years ago I grew exhibition carrots under glass with some success.
I grew them in forty gallon drums filled with river sand.
The river sand was drilled with a crowbar to form a cone shaped hole, 3"-4" diameter at the top..
This hole was filled with a very good sifted potting compost then thoroughly soaked.
Then I sowed my seed on top of the compost and lightly covered them.
Then the secret is in the watering, basically you dont want to water the hole, water the sand, the compost will extract the moisture from the wet sand by means of capillarity.
Too much water and your carrots will sweat and rot (crown rot) particularly if grown indoors.
The idea is let the carrots seek the water, in doing so they will get longer,if the water does not get down the barrel neither will the carrot.
I grow outdoors now where I do things slightly differently;
Up to the sowing stage the procedure is the same, I only keep the surface moist (manually during dry spells) until germination, then I dont bother watering them again, I let the weather do it for me, and strangely enough I get better carrots on a dryish year.
I hope this helps, so moisture is the key followed by ventilation to keep humidity down.
I hope that helps, and best of luck.
Just for the record, this is what mine looked like. As a guide the black tray is to the Garden News Tray' specification which is as I recall 24" x 18" so that makes theses carrots around two foot long and 3" diameter at the shoulders.
Ahhhhhhhhh! those were the days!!!