What is the nature of the transaction that gave rise to the cause of action that you are asking about? Is it with respect to real property, personal property, services provided, or something else? Was there a written contract? Since I don't usually hear laypeople ask about either unjust enrichment or accounts stated, I suspect that you have been served with a civil complaint and are curious as to what, if any, defenses you may be able to raise. The specific answer to your question will require more information to ascertain, but more generally I would encourage you to seek the assistance of an attorney. Trying to get narrow advice like this in order to represent yourself in defending a lawsuit is a little like asking a surgeon to mail you a scalpel so that you can perform surgery on yourself.
Any answer(s), and/or comment(s) provided are not legal advice, should not be relied on in lieu of consulting with an attorney, nor are they intended to create an attorney-client relationship. Those readers and / or posters who believe that they may have legal claim(s) should seek the advice of an attorney who is licensed to practice in the relevant jurisdiction.
Generally speaking, the statute of limitations on account stated or unjust enrichment is 4 years from the date the cause of action "accrued". When a cause of action "accrues", requires legal analysis of the specific facts that lead to the filing of the claim. Also, if the cause of action arises out of a written contract, the statute of limitations may be extended to 5 years.