That all depends on how good the soil is. the poorer the soil, the deeper you need to cultivate it by adding humus.
Most veg though grow in the top 6 to 9 inches(root wise) so most of the time you only need to cultivate that part. Just add lots and lots of humus. (If you can get it that is).

I agree with Owd and of course you can grow stump carrots, short roots so that isn't a problem.

Partly depends on which roots you are trying to grow. Carrots like sand, rather than humus, and the long ones, such as 'Scarlet Nantes', need deep soil. Short carrots,such as 'Royal Chantenay', 'Danvers', 'Little Finger', or 'Tonda di Parigi', are much less fussy. Beets like humus, but most are happy with just 20-30 cm of good soil. Potatoes and parsnips are the real hogs of the root world, best with around 60 cm of humus rich, soft earth. Turnips and swedes like to be treated like beets, but with a little extra lime in their diet, to ward off club root. Most radishes don't seem to care much about soil, though they may be stunted by hard clay. Daikon radishes are a little more fussy, wanting at least 30 cm of soft soil.

Thanks for your help, are going to try carrots and beetroot (also, know they are not root veg. but French beans