For one, there were about ten less bowls back then.
The next part I could be wrong about - I think the wording of bowl eligibility at the time was that a team had to have a winning record. Since there were 11 games at the time, normally six games would do it. But since Ohio State was 6-6 with an extra game, they weren't eligible. I know there were some changes implemented when 12-game seasons became the norm (for example, one win over an FCS school can be counted toward bowl eligibility EVERY year instead of every three or four years) and I think one of those was specifically wording it so that 6 wins is a minimum.
Again, I'm a little foggy on those details. I do know there were 6-5 teams that also did not make bowl games that year.