Translation:Were it not for him, I would have a big problem.
In this sentence "היתה" refers to "בעיה גדולה" which is third person feminine. Because of the "לי", it's 'I'. So -
I had a house = היה לי בית
I had a problem = היתה לי בעיה
She had a problem = היתה לה בעיה
If this is feminine, why is there a male voice in the recording. That really confused me. Or is הייתה used both for the feminine and masculine?
The word היתה in this sentence refers to בעיה, so it would be the same for a male or a female speaker. The next word, לי ("to me"), indicates it's I who had the problem, but it's the same for male or female.