Usually by looking to see how big they are. The beet at least should surely have been ready ages ago? Probably not going to grow any more now anyway. Carrots can be picked from being very small as people often pick the thinnings and use them in salds.

also you can cover carrots with a thick layer of straw during winter and pull them when you want them.making sure you put soil or such like on top to keep the straw from blowing away and for added insulation.good luck.

Carrots taste absolutely beautiful if they've had a frost on them - makes them really sweet. HOWEVER, when I have left them in the ground for any length of time I do tend to find that the slugs get at them below soil level, so it's a gamble.

Slugs will also attack beetroot under the soil level - you don't know it's happened until you lift them.

As for when to harvest beets, when the days get shorter and cooler, everything stops growing, so you might as well lift them. You could have lifted them weeks ago really. Other than that, you can just have a look and see how big the crown is - you might want great big beets or baby ones, it's up to you!