I’ve run out of holiday but I want to take some more time off this Christmas. Can I just do it? What’s the law on unpaid leave?

There are two main types of holiday to which you may be entitled: statutory annual leave and contractual annual leave, and unfortunately, once they're gone, they're gone. If you have used up all of your holiday entitlement, then any further leave can only be taken with your employer's permission and as you suggest, this is likely to be unpaid.