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.
Looking for content from the St. Louis Post-Dispatch? Articles from 1989 to the present are available online through ProQuest's Newstand. Some libraries have archives of older issues (usually on microfilm), which can be searched by date and requested through Interlibrary Loan.