Should onions be kept in fridge?

It is not necessary to keep onions in the fridge - they should basically keep fine in any cool, dark place (e.g. on a rack in a cupboard). However keeping them in the fridge will certainly not harm them, and it does have some advantages - it will help to retard sprouting of the onions, and also retard mold growth; also keeping the onions refrigerated will reduce the 'tearing' effect when you cut them. Indeed, popping them into the freezer for a few minutes before you cut them will help even more to stop them making your eyes water.