Buckets are a fantastic option of container to grow carrots in as they are deep, a 5 gallon bucket tends to be around 12-15 inches tall and a 10 gallon bucket between 16-18 inches tall. As with growing any plants in containers, make sure there are always holes in the bottom of a planter to aid drainage.
The number of carrots you can grow in a bucket will vary, it will depend on the variety of carrot you are growing and whether you want to harvest them as baby carrots or let them grow to full size. A 5 gallon bucket is likely to be able to hold around a dozen carrots, while a 10 gallon bucket can house around 24 carrots.