If you want to store onions after you cut then you should put the cut up or cut onion or onions into a sealed container and then place the sealed container with the onions in it into the refrigerator.
If you store the cut onions in the refrigerator in the sealed container the cut onions will last up to 7 days in the refrigerator before going bad and needing thrown out.
After 7 days the onions should be thrown out because they could make you sick.