You can arbitrate a lot of legal disputes, but not all of them. For example, you cannot arbitrate a homicide. If it is a criminal dispute, it is heard in court.
You cannot arbitrate whether you are married, but you could settle child custody and financial issues through arbitration. In fact, a married couple can avoid high legal fees if they both agree to arbitration to settle the finer points of their divorce.
You can arbitrate other civil law issues, such as contract disputes between a company and a contractor.