I'm attempting to obtain payroll records from the state of California for the year 1989 for the purposes of (hopefully) calculating the number of hours I worked. However, I filed exempt for fed and state income tax because I was a student at the time. I'm hoping I can use the OASDI/EE tax to do it, but I don't know if this is possible. Does anyone know how I may go about obtaining those payroll records?

You are going back 24 years. I am not even sure California would retain the records for that long.

I suggest you call the Franchise Tax Board and hope they have a process to allow you to get that information.