I have looked for rat droppings but can find none. My instinct would be to say two-legged thieves, but the apples were wizened and a bit soft, not an attractive eat, good for cooking but not eating raw - plus who on earth knew about them and could reach them? It is certainly not DH, he is a fussy eater and wouldn't touch them.
The shed concerned is 100 feet plus down a garden which is surrounded by a six foot fence in good order, the overgrown orchard garden of older neighbours wraps round two sides of the garden and they do not even harvest their own apples and the other neighbours are a young family, whom we know well, who again have apples galore of their own and children too young to forage. The garden can only be accessed through the house or via a narrow side entrance with a six foot locked gate.
rosesare red I think you are right it is squirrels. We do have them in the garden as we have a huge hazel tree. The shed door does not close well so it would be easy for them to get in. The apples were in a plastic box on a chair. Difficult for mice to access, the usual problem, but not a squirrel. However they did not touch the carrots stored in adjacent box, although I think they may have disturbed the lid.
I need to invite Sherlock Holmes in to solve the mystery