Oh wow, I'm sorry you have had to go through that, especially with family who you would think would try to be as supportive and motivating as possible. You deserve to have them in your corner and I think you should let them know how you feel.
I don't think it was appropriate for your sister to tell you what her friend said. Not only was it down right mean but if she is concerned for your health then she should address you as her sister in a loving, helpful manner. That goes for your father too. There are many here that will tell you that an unsupportive family just made them even more rebellious and wanting to eat. Some feel it is the only control they have in their lives whether they eat or not.
Yes, unfortunately, we are known as the land of the disposable and fast food. Many of us struggle with it. Coming on here is a good step, try to find others around you that will support you if you can't get it from your family. You need to have positive messages going in your mind to offset the negative. Try to think why YOU want to do this. What is your motivation? All the nagging in the world isn't going to keep you going in the right direction if you're not ready.
When you are, you don't need to starve yourself, just look for healthier alternatives to those foods you love, keep active
and plan, plan, plan. The traps of fast food are that it's cheap, fast and easy. You're going to have to plan and prepare for those times you need something quick.
Keep looking for that support and speak up for yourself, you deserve it! All the best to you.