Radishes: the one that works easily
Radishes are the exception. They germinate in three to five days, mature in 25–35 days, and a small root crop suits a net pot perfectly. They are also useful as a marker crop — sow them among slower seeds and they will show you where the rows are before anything else is up.
They bolt and go woody in heat, so grow them in cooler conditions and do not leave them in past their date. Sow successionally every two to three weeks for a continuous supply.
Carrots, and why they fork
A carrot that forks has met something. A stone, a lump of clay, fresh manure, or the wall of a pot. In a net pot filled with clay pebbles the roots will meet pebbles constantly and the result is a tangle rather than a carrot.
If you want to try, use a deep pot with a finer medium — a coir and perlite mix rather than pebbles — and choose a short, stubby variety bred for shallow containers rather than a long tapering one.
Carrots also need consistent moisture and low nitrogen. Too much nitrogen gives a huge leafy top and a tiny root.
Beetroot
Beetroot is the most tolerant of the root crops in a container. It germinates from a seed cluster that produces two or three seedlings, which need thinning to one — that is the main thing people get wrong, and the reason for small or misshapen roots.
They want pH 6.0–6.8 and moderate feeding, EC 1.4–1.8. Too much nitrogen gives leaf at the expense of root.
Beetroot is a biennial, so saving seed means overwintering the root — the same process as a brassica, and like chard it crosses with anything else in Beta vulgaris.
Saving seed from biennials
Carrot, beetroot and turnip are all biennials: the root you harvest is year one, and seed comes from the same plant in year two after a cold period. That means selecting a good root, storing or replanting it over winter, and letting it flower the following spring.
All three are insect-pollinated and cross readily. Carrot will cross with other carrots and with wild carrot, which is a common weed and will ruin a seed crop quietly. Beetroot crosses with chard and with leaf beet. Turnip crosses with pak choi and mizuna.
Isolation distances are in the hundreds of metres for all of them. Radish is an annual and much easier — it sets seed in the same season, in long pods.