They can be in open trays but the crucial thing is that they have to be in the dark otherwise they go green and nasty - cupboard under the stairs? Cupboard in garage/shed?
... and don't wash them... just brush off dried dirt. We use burlap sacks for storage. I think it allows the air to circulate a little better... but concur with everyone else suggesting dark and cool...
You can buy a black lined potato bag from Poundland: it pulls tight via a drawstring, and you can reach said spuds from a zipper near the base of bag. It does the trick with shop bought spuds, so I cannot see a reason why it should not with home grown.
I keep mine under a shelf unit in a cool porch. In bags (5kg sugar bags) inside a cardboard box. Small bags so that if some rot from blight they won't all go.
I used to store things in the garage nut rodents made a right mess of the apples a couple of times, so I don't now.