What should I do if I want to take an ESS class that is restricted to majors but I am not an ESS major?

The restriction will be lifted during the adjustment period (usually about a week before classes begin for the quarter).  At that time, you should be able to register, unless the class is already full.  If that is the case, you need to speak with the instructor directly to see if he/she will be taking more students.  For specific dates of the adjustment period, consult the appropriate academic calendar.