I was going to vote 2003 on the basis that it felt to me like Ohio State was more "in the running" that year than they were in 1997 but upon further review I'm not sure that is correct.
Ohio State was knocked out of the running by losing to Michigan both years. Also, Ohio State was #4 when they lost to Michigan both years.
Initially I thought that Ohio State was more realistically "in the running" in 2003 but that is mostly because they were ranked higher pre-season in 2003 (due largely to being the defending Champion).
In 1997 Ohio State started out #9 and climbed to #7 before losing to PSU on October 11 and dropping to #11. Then they climbed back up to #4 before THE GAME.
In 2003 Ohio State started out #2 and drifted between #2 and #5 until losing to Wisconsin on October 11 and dropping to #8. Then they climbed back up to #4 before THE GAME.
In retrospect, realistically Ohio State was closer to the NC in 1997 than they were in 2003:
When Ohio State played Michigan on November 22, 1997 the Buckeyes were #4 and the Wolverines were #1. #2 FSU subsequently lost to Florida. That and winning the Rose Bowl* theoretically would have gotten the Buckeyes up to #2 and they just would have needed one upset of Nebraska to win an NC.
*Rose Bowl. At the time I thought that Ohio State had no chance whatsoever at the Rose Bowl. Had the Buckeyes beaten Michigan then the Buckeyes and Wolverines would both have finished 7-1 in conference. At the time I (and everyone else) assumed that PSU would easily beat MSU which would have created a three-way tie where tOSU>M>PSU>tOSU. At the time the Big11Ten broke such ties with the "longest loser rule" under which the team that hadn't been to the Rose Bowl in the longest got the bid. In 1997 that was Michigan because tOSU went after the 1996 season and PSU went after the 1994 season. However, the Nittany Lions lost to MSU so if the Buckeyes had defeated the Wolverines the Buckeyes would have gone to the Rose Bowl on the basis of their tie-breaking H2H win over Michigan.
When Ohio State played Michigan on November 22, 2003 the Buckeyes were #4 and the Wolverines were #5. Thus, even with a win the Buckeyes would still have needed at least two upsets just to get to the BCSNCG and they would have had to win that as well.
Thus, I voted 1997.