The closest Presidential Elections in American history would be the following in chronological order since the introduction of popular vote in 1824:
Presidential Election of 1824—Andrew Jackson vs John Quincy Adams, Henry Clay, and William Crawford
Presidential Election of 1876–Rutherford B. Hayes vs Samuel Tilden
Presidential Election of 1880–James A. Garfield vs Winfield Scott Hancock
Presidential Election of 1884–Grover Cleveland vs James G. Blaine
Presidential Election Of 1888–Benjamin Harrison vs Grover Cleveland
Presidential Election of 1892–Grover Cleveland vs Benjamin Harrison, James Weaver
Presidential Election of 1916–Woodrow Wilson vs Charles Evans Hughes
Presidential Election Of 1960–John F. Kennedy vs Richard Nixon
Presidential Election of 1976–Jimmy Carter vs Gerald Ford
Presidential Election of 2000–George W. Bush vs Al Gore, Ralph Nader, Pat Buchanan
Presidential Election of 2004–George W. Bush vs John Kerry