Feelings of which, at sixty-odd, I felt ashamed.
What is sixty-odd?
It means that the speaker is in his sixties (ie his age falls somewhere between 60 and 69). When people don't want to use a specific age, or don't want to say exactly how old they are, they use this kind of phrase.
How old are you?
Oh, forty-odd!
We also say "forty-something".