Archives exist in many places. They are often part of academic institutions, but other archives are units within public libraries, historical societies, museums, and government offices. Some companies have archives as well.
Even though archives are often part of library systems, you do not have to be a student at a particular university or college to use that institution's archives. The exception is business archives; some corporations prefer to limit access to records to company employees. It's always a good idea to do a little reserach before you visit a facility and make an appointment if you're interested in something specific or want the help of the reference staff.