All Records in Salesforce must have an Owner which is either a User or a Queue, if you try and set OwnerId to Null this will always fail as the field is mandatory therefore Null is not an acceptable value.
With the records that you have set the ownerid to "" - what actually displays on the record?
I am not sure I understand what business case there is for not having an Owner for a record? Even if it is not an individual then a queue should be used for the record to be picked up from for update.