The short answer is that an urban planter box for carrots should provide at least 12 inches (30 cm) of usable soil depth, and 15–18 inches (38–45 cm) is even better if you want to grow longer carrot varieties. Carrots are root crops, so the soil depth directly affects how straight, long, and well-formed the roots can become. A shallow box often leads to short, stubby, twisted, or forked carrots, especially when the soil is compacted or contains stones.
If your planter box sits on concrete, pavement, or a balcony, measure the actual soil depth inside the box rather than the outside height. Do not count a drainage layer of gravel as usable soil depth. For short or round carrot types, an 8–10 inch (20–25 cm) depth can work, but 12 inches is still a safer minimum for most urban gardens. For medium carrots, aim for 12–15 inches. For long carrots such as Nantes or Imperator types, choose a box that gives you 15–18 inches of loose, stone-free soil.
The soil mix matters as much as the depth. Use a light, well-drained mix that is free of large stones and heavy clay, and avoid fresh manure that can cause forking. Make sure the planter box has drainage holes, because carrots rot in waterlogged soil. Keep the soil evenly moist during germination and root development, and thin the seedlings so the roots have room to grow. In short, 12 inches is the practical minimum, 15–18 inches is ideal for most carrots, and shallower boxes should be reserved for compact or round varieties.