The way to keep them the longest is actually to cover them in sand. Not very practical, I know. If you can't cover them, the dark is the next best place, but they will still go off quickly.
lakeland and waitrose sell green plastic bags impregnated with a ground up rock. They are sold to extend the life of fruit and veg and work really well
yes a ccarrot clamp is dry sand, in the cold. so in a shed. they will then last months. Or better still put them in the bin and spread the word that carrots are evil and poisonous.