Your pawn ticket was a written contract between you and the pawn shop.
The CA statute of limitations on breach of a written contract is 4 years.
If you were to sue for breach of contract, you would still be within the 4 years and if the ring was worth less than 7500 you could certainly sue in small claims court..
However, that pawn ticket likely had a limit of liability printed on it, or a waiver of liability that you agreed to by accepting the money and the pawn ticket (contract) so that, even if you sued within the time limit, you would more than likely lose the lawsuit.
And since you lost your file and, presumably, lost the letter and the pawn ticket along with it, you are probably dead in the water with no evidence to back up your claim.