I've heard so much trouble about growing carrots in containers; in fact most root crops have a hard time.
Carrots have a main tap root and it needs friable soil with no impediments such as lumps of clay, unmatured compost or rocks to mess with.
Surprisingly carrots also have a mass of small fibrous root hairs which go out and down a long way. Some of these root hairs are so small we don't see them when pulling out carrots, but the general rule is to allow about 60cm (2ft) so that the roots can get the sustenance to grow.
If you space carrots only a few cms apart you will need to constantly feed and water them, whereas at 15-20cm (6-8") apart they will have a better chance of growing to maturity in a container. ~ Megan ~