First of all, the Validation Rule executes upon a DML event on an object. For argument's sake, we'll assume that object is Obj1__c

So, as long as Obj2__c is a parent object (via master-detail or lookup) to Obj1__c, then the Validation Rule would look like

Enter_Required_Quantity__c > Obj2__r.Quantity_Left__c

Obj2__r is the relationship name from Obj1__c to Obj2__c

Now, if Obj2__c is not related to Obj1__c via a master or lookup relationship (parent/grandparent/great-grandparent or greatgreat-grandparent), then you can't do the Validation Rule.

Validation Rules can't look down to children nor look up and then down to siblings.

It seems that your fields Billing_Period_Start__c and Billing_Period_End__c are of type Date, hence the error. Remove the DATEVALUE() function from your formula and you should be good to go.

As mentioned in the DATEVALUE() function documentation

DATEVALUE() - Returns a date value for a date/time or text expression.

Its worth noting and reading the tips section of the article given above, out of which, two are related to your issue

tips

• If the field referenced in the function is not a valid text or date/time field, the formula field displays #ERROR!
• When entering a date, surround the date with quotes and use the following format: YYYY-MM-DD, that is, a four-digit year, two-digit month, and two-digit day.