This sentence, as written here doesn't make sense to me as a native British English speaker:
You can have feminist ideals and, as a woman, support yourself, but that if you do that you won't make as much money as you would if you married a rich man.
The extra "that" which is the subject of your question just shouldn't be there, if this sentence is complete and is a statement.
I'm going to add a bit more to your sentence and make it make sense without taking anything away:
He said that you can have feminist ideals and, as a woman, support yourself, but that if you do that you won't make as much money as you would if you married a rich man.
Now your extra "that" makes some sense. In this sentence the narrator is quoting someone, without quoting verbatim. When you say "He said that..." as opposed to "He said" it is understood that what follows is a summary, or the essence of what was said. If you quote someone verbatim, you use quotation marks.
So the meaning of my sentence is that "he" (whoever he is) said that you can have feminist ideals and, as a woman, support yourself, but he also said that if you do that ("that" being what he previously said) you won't make as much money as you would if you married a rich man.
EDIT
You have since cited the full quotation, and I was basically correct - there was more that preceded the sentence you typed, and it was a summarised quotation.
The short interpretation of the sentence is that she's saying you can have feminist ideals and, as a woman, support yourself - but that if you do that you won't make as much money as you would if you married a rich man.
The writer here even tells you that what he is about to say is a "short interpretation" of something somebody else said, and he summarises it by saying that she said two things:
- That you can have feminist ideals and, as a woman, support yourself
- That if you do (have feminist ideals and as a woman support yourself), you won't make as much money as you would if you married a rich man.
The second "that" still doesn't strictly need to be there, but the person quoting uses it to show that he still summarising the opinion of someone else.