First of all, you're a 35 year old woman. You should be entitled to your own decisions. Maybe on a side note, you should be assertive and tell your family that you have every right to go to that interview. They shouldn't and couldn't forbid you for doing so.
Anyway, if you really want to go to that interview, why not? That interview might give you good opportunities. If you're planning to go to the interview, either have your car fixed before the interview or look for alternative transport (like public transport or uber). If your family is no help, call a friend.
I hope you good luck on your endeavors and to that interview!