This year I planted 10cm apart in rows 30cm apart. This meant the plants in the row were very close together but I was able to inter-sow both late carrots and beetroot between the rows.
The beetroot didn’t really develop, but I’d sown direct. The carrots which I’d done in small seed trays have been a great success. They’ve given me an extra crop of small (dipping I mayonnaise size) to medium size (steamed and eaten whole). They were resistafly and an early nantes. That means that I’ve not had to pull that many of my earlier planted ones so I’ve got more for winter storage.