A number of research databases are available to Ohio residents with a public library card. See:
Ohio Web Library: http://www.oplin.org/databases/
Many other states also have statewide agencies that purchase databases for all residents, including Michigan, Minnesota, Missouri, etc.
Most of the library's databases are only available with a subscription (and are very expensive), which is why only BGSU students, faculty and staff can log in and use them from off campus. However, many research databases, particularly those produced by the government, are free. This list will help you determine if a database you use frequently can be accessed for free.
Here's a selection, by subject:
Art & Art History
Biography
Business
Criminal Justice
Dissertations
E-Books
Economics
Education
Ethnic Studies
Film
Law & Government
Library Science
Medicine, Health & Nursing
Music
Sciences
Social Sciences