しない is a negative conjugation of する
And from what I understand でもなかった is the equivalent of "was not"
So it's a double denial? How does this work?
Yes, しないでもない (≒しなくもない） is a double-negative construction, which can be seen as a set phrase used to say something less confidently, or to reluctantly admit something.
(lit.) It was not that I didn't feel I came to be able to read his character.
I feel I have gradually learnt how to understand his character.
- 失敗だったと言えなくもない。 ≒ 失敗だったと言えないでもない。
(lit.) It is not that one cannot say it was a failure.
One could argue that it was a failure.
- 100万円はなくもない。 ≒ 100万円はないでもない。
(lit.) It is not that I don't have one million yen.
(Admittedly,) I have one million yen.