If you are saying something like that his career was very average, then you would be better to say something like: "My father thought that he didn't either succeed or fail in his career." I would add the 'either' to make it clearer, change the 'on' to 'in', and I've corrected the spelling of career.

My father believed **that** he didn't succeed or fail **in** his carrer.

Your statement is contradictory. You are saying he didn't succeed, and you are saying he didn't fail at the same time.

My father thought he didn't succeed in his career. My father thought he failed at his career.

Actually this sentence got nothing to do with a father. It is about writing. Writing = literature. Succeed or fail on strength to persuade = whether it convinces well or poorly. Does not fail = does do, acts well, acts even better, does a good job is good is really good regardless of its ability to convince. Your sentence is also good, but there it is "succeed in" not "succeed on", because one succeeds ***in*** career, NOT ***on*** career. Whether my dad got promoted or told off by his superior year in year out, he did not fail to believe himself as success in his career.