Doubt from people close to you often sounds like a judgement and comes from fear of watching you get hurt. They are usually not arguing with your ability, they are imagining what happens if it does not work and they cannot fix it. Knowing where it comes from does not make it feel better, but it changes what you are actually arguing with.

Instead of defending the whole plan, tell them the next step and what you will do if it fails. Update them when something goes right. People who are frightened calm down with information, slowly.