Use a cup at least 4 to 5 inches tall to give the carrots room to grow, and use wider cups for more planting. Line the bottom of the cup with 1/2 inch of pea gravel to ensure drainage in the cup. Carrots rot and die in standing water. Fill the cup full of organic compost, garden loam or a mixture of the two to give the carrots a loose, moisture-retaining base. Add 5-10-10 fertilizer for more nutrition.
Plant a small carrot cultivar such as Baby Spike, Little Finger, Minicor, Short 'n Sweet, Orbit or Thumbelina for smaller container growth. Push the seeds 1/2 inch into the soil with two to three seeds in every inch of space.
Water the soil with 2 inches of water from a watering can, to avoid displacing your seeds, and put the cup in a spot with four to six hours of sun every day and temperatures of 55 to 70 degrees Fahrenheit. Carrots don't like hot, dry temperatures or sun.
Maintain soil moisture with 2 inches of water every four to five days. Carrots won't grow without adequate moisture. Thin the seedlings to every inch in the cup when they sprout. Harvest the carrots when they reach 3 to 4 inches in length or at their maturity date.