Public Libraries:
When you find a citation to the article you need and you can't access the full text - or you hit a paywall, you can often use your local public library to acquire the journal. While it's much more common for books, most public libraries will interlibrary loan articles for you as well.
University Libraries:
If there is a public university near you, you might also check to see if there are computers dedicated to public research. Most public university libraries offer access to their electronic resources through a select number of computers within the library. UVM's Howe Library offers this type of access via public research computers at the front of the building.