Carrots are an ideal vegetable crop for container gardening. While you may not grow some of the giants you have seen at your supermarket or in competitions, many carrot varieties grow very well in limited space. Small varieties that grow in ball shapes, such as Thumbelinas, are obvious choices. Plant in a pot at least 1 foot deep to expand your carrot-growing options to include most varieties currently available for planting. Carrots prefer a rich soil high in organic matter, which is much easier to control in containers than in the ground.
Mix potting soil and compost together in a 1:1 ratio. Do not use soil from your garden or yard, as it could contain soil-borne diseases or insects that might harm your carrot seedlings.
Mix 1/3 peat moss or coir into the potting soil/compost mix. Be aware that there is an ongoing controversy over whether or not harvesting peat moss is harmful to the environment. See The "Carnivorous Plant FAQ" link below for details with which to make an informed decision.
Sow carrot seeds at a depth of about 1/2 inch. Sow as many as you like, as close together as you like. Keep in mind, however, that you will need to thin out the seedlings once they reach a certain height. Overseeding helps protect against those few seeds that inevitably do not germinate, but grossly overseeding will just waste your precious seeds.
Cover over the seeds lightly with soil. Water using either a watering can or a garden hose with an adjustable sprayer attachment. Choose a gentle spray setting, if using the latter. Settings labeled "mist" or "shower" usually work well. Do not choose a forceful spray, as this will displace seeds and soil. Seeds will sprout in about 7 to 14 days.
Apply a good all-purpose vegetable fertilizer according to package instructions. This fertilizer should be applied approximately once every week to two weeks once carrots have established themselves.
Thin out your seedlings to one per half-inch when they reach 1 inch in height. Re-pot or use the seedlings you remove. All parts of the carrot plant are edible for humans, and many animals eat them too--rabbits especially love the tops, so save them for any pet bunnies you may have.
Water regularly. Carrots require consistent watering in order to stay strong and resist diseases and pests. Containers dry out more quickly than in-ground gardens, so you will need to monitor the water situation carefully. Consider buying a moisture meter to keep in the pot. This handy tool will help you to avoid over- or under-watering your carrots.