Flip the barrel upside down and inspect it for drainage holes. If no holes are present, drill four to six ½-inch diameter holes in the bottom of the barrel. Flip the barrel right side up.
Fill the barrel with moistened potting soil to within 3 inches of the rim. Sprinkle ¾ cup of slow-release fertilizer in a balanced formulation such as 14-14-14 on top the soil and mix it in with the soil in the barrel.
Sow the carrot seeds on top of the soil surface. Plant approximately 1 seed per 2-inch square. Cover the seeds with a 1/8-inch layer of moistened soil.
Stretch plastic wrap over the top of the barrel or set a sheet of glass on top of the barrel. The plastic or glass helps keep the soil moist during germination. Water the seeds only if the soil surface begins to dry completely.
Remove the plastic or glass once the seeds sprout, approximately two weeks after sowing. Thin out any extra carrot seedlings so the remainder are spaced at least 2 inches apart.
Check the moisture in the barrel daily. Water the carrots when the top 1/2 inch of soil feels dry. Irrigate until water drips from the bottom to ensure all the soil in the barrel is evenly moistened.
Harvest the carrots approximately two months after they germinate. Pull up one or two carrots. If they are the desired size and maturity, harvest the remainder.