I will start by briefly explaining what 「お見合{みあ}い」 is for those unfamiliar with the custom and/or the term. It is a formal kind of meeting for a man and woman with a view to marriage. The conversation above is taking place at such a meeting.
From the convo, 中野{なかの} is the man and 大和{やまと}, the woman. As anyone would expect, "hobbies" is a common topic at お見合い and that is exactly what is being talked about here.
As seen in the convo, however, the two people might not share a favorite pastime. So, the woman says:
「私がしたことのないことばかりです。」
= "I have no experience in doing any of those things."
This is said upon finding out what the man's hobbies are.
The first こと means "experience" (～ことがある/～ことがない) and the second こと means "things". Thus, using こと twice is not at all redundant. The sentence is perfectly correct and natural-sounding.
Finally, could 大和 have instead said:
「私がしなかったことばかりです。」?
No, no chance at all. Why not?
The two are discussing their hobbies. They are not discussing the actions they performed on a particular occasion in the past. They are talking about what they like to do habitually. This is why you cannot use the past tense 「した」 or 「しなかった」. Hope you are following this logic.
Let us suppose that you went to Hawaii last year and I went there just recently and we are talking about what we did while in Hawaii. You did activities A, B, C and D and I did X, Y and Z.
Upon finding out about what I did, you might say:
「私がしなかったことばかりです。」
= "I didn't do any of those things." Or more literally, "All of those are the things that I didn't do."
That will be a very natural reply.
