In the story, Nene is a teacher at a school for girls in Lagos.
We find this out during Nnaemeka's conversation with his father, Okeke, when he is breaking the news that he is engaged to Nene.
Unfortunately for Nnaemeka and Nene, Okeke is extremely opposed to the idea of women teaching. He abides strictly by Biblical beliefs, and he sincerely adheres to the notion that women shouldn't teach others and, further, that they should "keep silent." Okeke quotes "St. Paul in his letter to the Corinthians," a passage from the Bible, when he reminds his son about his views.
In fact, Okeke doesn't just disown his son and leave it at that--he feels an ongoing struggle and deep shame about the fact that Nnaemeka went forward in a marriage to a woman who teaches. Okeke has other problems with Nene (mostly because she doesn't belong to the Ibo tribe, and because she wasn't part of a traditional arranged marriage) but Nene's profession is definitely a factor that prevents Okeke from accepting her as his daughter-in-law.
You might ask, what kind of job should a girl have in order for her to be accepted by Okeke as a suitable wife for his son? We get a clue in Nnaemeka's description of Ugoye, the traditional girl he was supposed to marry but didn't love:
"When she stopped schooling some years ago her father (a man of sound judgment) sent her to live in the house of a pastor where she has received all the training a wife could need. Her Sunday school teacher has told me that she reads her Bible very fluently."
So really, Okeke would prefer a girl who has no profession or even schooling. He would prefer a daughter-in-law who simply reads and abides by the Bible.