My employer has verbally told all employees that we are not allowed to bring any personal items to our workplace such as a purse, cell phone, beverage

Or food. We are only allowed to bring our car keys and they must be locked up. We are not provided with a break room or lockers or anywhere in our building to put our personal items. Is this legal? It''s not in writing anywhere. They claim its because they have had employee theft problems at other locations. We are on video monitoring at all times.