Ask HN: Path to become a Product Manager?

Background: I'm 21, live in the UK, without a degree, and split my time between building simple Wordpress websites for freelance clients, working in retail, and slowly learning to build web apps using Ruby on Rails.

My 'aspirational job' would be to work as a Product Manager at a startup, wrangling something into existence and using data to refine it into something better.

I can't seem to connect the dots between there and here - I don't have the experience to get a job now as evidenced by many tens of rejected applications, and there are very few 'junior' roles in London to learn the craft.

One avenue I'd considered is to try and find a non-tech position in a startup, and try and learn from their product team if the role permits.

Has anyone been in a similar situation, that has any advice to share? I have the drive to get there but can't see quite which route to take.