There's no problem with deeper boxes, but you'll be putting a lot more weight on the balcony, and you'll be spending more on soil. I had a school garden about 6 inches deep that was on asphalt, so the roots could only grow in the box. We didn't have long carrots, but we grew broccoli, strawberries, and tomatoes, plus marigolds and mini sunflowers for color. You can compromise and go with 10 inch deep boxes.

Here is a link that might be useful: What's Growing On?

I like more soil that the 6 inches, but as the teacher stated, you will be adding a LOT more weight. As for Carrots, try the shorter varieties i.e. Chantenay Red Core or Parisienne. Source; Baker Creek Heirloom seeds. http://rareseeds.com/ Lots of good stuff there.

In My SFG book on page 65, copyright 1981, Mel said 12 inches for poor soil & 6 inches for average soil.
I away double dig(16-24 inches) my beds (I know OGM now say you do not have to do it), but I am old school & know deep digging one time when you are making the bed is better then wasting seed, time, water.
Would you leave your pet in a hail storm with out shelter
I say do it right the first time & you will know it is not the hard soil when your garden is slow to grow or produce.

6" works fine. The roots spread sideways. I wouldn't have believed it either, but I was amazed at how my tomatoes went out, not down.

j

yep, the roots are all over the garden. kind of amazing.

i personally went 12in high and set mine down in the earth about 4in(more for temperature, at least in my mind)

that said the only time ive really used the full depth is for potatoes and maybe the carrots. also, it helps for trellis/tomato stake. i bury a bunch of stuff in the fall (last of the summers compost, and leaves/old plants from the garden).maybe it helps out every spring when i turn the soil over, seems the stuff on the bottom is in better shape, that may be in my head though

something else i notice about it being a little higher, the dog stays out of it, and it forces people to be mindful of it as well. and again probably in my mind, but ive never had rabbits wreaking havoc in the garden either. i doubt if one has anything to do with the other.