Clever balancing act?
Grow? Yes - but only, I believe, to their 'design size'. Otherwise everyone would have record-breakers? Obviously they are nicer younger but, as with a courgette, an 'unripe' fruit will not keep. They like to be exposed to sunshine to ripen, Preferably on a slate - or somesuch - to keep their bottoms dry.
Quote from BBC Gardening - "If you want to store marrows, wait until the stem is just starting to dry out, at which point the skin will be quite tough. Then cut them with a long stem. They store best at 7.5°C to 10°C (45°F to 50°F)." They should then keep 3 months or so. = Tim
PS The record for a marrow is 102lb.