Some vegetables work well in containers and some don’t. Sometimes there are even advantages of growing vegetables in pots or other containers! Carrots can go either way – it’s important to do them right.
Same as if you are planting them in the garden: you are growing carrots for the ROOT, and it is a long root. So that is your prime priority!

If you are using a quality potting soil, you do not have to worry about that tap root hitting a rock or any other obstruction. But make sure that your pot is deep.
And select a seed type for a shorter carrot variety (the one known as Paris Market is so short it’s shaped more like a radish!) I also like the variety known as Bolero – both short, and cold resistant for winter harvest.
And – same as for your general garden – they are tiny seeds. So sow them very shallowly (just a fine sprinkle of soil on top) and keep them well watered for the first 10 days, as they send down their tap root.
Don’t sow the seeds too thickly. It’s only a small pot, so you can be careful how you sprinkle. Try to get them around 1/4 to 1/2 inch (or .5 to 1 cm) apart when you sow. Once they are an inch or so high, then thin them to a minimum of an inch (or 2.5 cm) apart – otherwise you can’t expect your carrots to grow very big! The farther apart they are, the more nutrients they will receive – so it is a fine balance between how many plants you can cram into your pot and how big they will ever grow for you. I find that having little clumps of two or three plants an inch apart from one another, then a bit more space between the next clump, is my most productive balance.

You are not gardening for market (where you want everything ready to harvest at once) – you are gardening for your household. Right?
You just want a handful of carrots – or maybe two – every week.
So don’t plant too many at once!
Sow one larger pot earlier in the spring, to get you through the first part of the summer. Then do numerous smaller pots, sowed every two to three weeks, which will guarantee you a harvest through the summer and fall and even through the winter.

Carrots actually keep really well in pots – even down to several degrees below freezing. Unlike carrots that you overwinter in your garden, which are susceptible to attack by wireworms, carrots in pots usually stay pretty clean.
I keep my carrot pots up on my balcony through autumn and the first half of winter. I may supplement with store-bought carrots to use for cooking – but these are my go-to supply for tastier and sweeter and crispier carrots to use in salads – better than anything money could ever buy.
I don’t have greenhouses or an “acreage” – but I still am able to provide a lot of food for my household from my backyard gardens and from my balcony here in town. If you would like to learn how you can become self-sufficient and grow your own food, too, then sign up for the Food Garden Club! It’s free – and I promise I will never spam you or share your private contact info.
I grow the little round Paris Market carrots in pots, after not getting v good results growing directly in my clay soil. I agree with you, there’s nothing like the taste of home-grown carrots, and I wish I had followed the advice you give and sown more pots to see us through winter (here in Belgium). There’s always next year, I suppose!
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