This page has the numbers for carrots: 3 inches between plants with 12 inches between rows, sixteen to a square foot, direct sown three weeks before your last frost, an inch of water a week, and a fifth to a third of a pound per root in 60 to 80 days.
Carrots need one thing the numbers do not capture: twelve inches of loose, stone-free soil with nothing rich added. The three sections below cover that, the two to three weeks of damp surface that germination needs, and the thinning step that is skipped more often than any other in the garden.
Soil first
A carrot is a taproot trying to go straight down. Anything in the way makes it fork, twist or stop. What it wants is loose, stone-free soil worked to at least twelve inches, ideally deeper.
What it does not want is richness. Fresh manure and nitrogen-heavy fertilizer both cause branched, hairy roots — carrots are one of the few crops where adding compost right before sowing actively hurts. Improve the bed the season before, then sow into settled soil. Phosphorus and potassium are fine; nitrogen is the problem.
If your ground is stony or shallow, do not fight it. A raised bed or a deep container of loose mix will give better carrots than any amount of effort on unsuitable ground, and a short, blunt-rooted variety will manage where a long one cannot.
Sowing
Direct sow about three weeks before your last frost date, and again every few weeks for a continuous supply. Carrots do not transplant — the taproot is the crop, and disturbing it defeats the point.
Sow thinly, about half an inch deep, and firm the soil gently over the drill. Then the hard part: keeping the surface damp for the two to three weeks germination takes. A board laid over the row, or a strip of fleece, holds moisture and makes the difference between a row and a disappointment. Check daily and water gently with a fine rose.
Radish seed sown along the same drill comes up in days, marks the row while you wait, and is harvested before the carrots need the space.
Thinning and spacing
Thin to three inches apart once seedlings are an inch or so tall, with 12 inches between rows. In a square-foot bed, sixteen carrots to a square is the right density.
Thin in the evening, in damp weather, and remove the thinnings from the bed rather than leaving them. Crushed carrot foliage smells strongly and that smell is exactly what carrot rust fly navigates by.
Water and care
An inch a week, steadily. Irregular watering causes splitting: a dry spell followed by a soaking makes the root swell faster than its skin can grow. Mulching between rows once the plants are established evens things out and keeps the crowns covered.
Draw a little soil over the shoulders as the roots swell to prevent green tops.
Harvest and storage
Pull a test carrot at about sixty days to see how they are doing. Loosen the soil with a fork before pulling in heavy ground, or you will snap them.
In cool weather carrots hold in the ground for weeks and get sweeter after a frost, which makes them one of the best crops to leave in place and harvest as needed. Where the ground freezes hard, a thick mulch extends that considerably. For storage, twist the tops off — leaving them on draws moisture out of the root — and keep them cold and damp.