More specifically, I need the sports sections. I'm doing an assignment for school on NFL history, and it would make my project 100x easier if I could see what journalists who were actually watching these guys had to say about certain players and teams.
I'd prefer that it be free but I'm not opposed to paying if I have to.
When I was running a Call of Cthulhu campaign I learned that all of the Boston Globe's papers are available (back to the 1600s!) for free on their website. I was able to give my players print outs of the actual day's news from the 1920s and tie in news stories into the eldritch mystery they were solving.
Chronicling America. It is a project of the Library of Congress but I believe it only goes to 1963. It is the only free newspaper site that I am aware of.
You may want to check your local (or, if this is a college project, school) library—many subscribe to services like newspapers.com, Proquest Historical Newspapers, or specific online newspaper archives like those for the NYT or the Times. Of course, any of these services could theoretically be accessed with an individual subscription as well, but why buy the cow when you can get the milk for free?