Firearms Owner Privacy Act of 1998 - Amends the Federal criminal code to prohibit an officer, employee, contractor, consultant, or agent of the United States, including a State or local employee or officer acting on behalf of the United States, who receives information provided to or derived from the national instant criminal background check system about an individual seeking to obtain a firearm, from knowingly retaining or transferring that information: (1) after 24 hours if the individual is not prohibited from receiving a firearm; or (2) after five years if the individual is prohibited from receiving a firearm (with an exception for information about an individual if a criminal prosecution has been commenced against the individual on the basis of that information).