I know how you're feeling. I want to work in the tv/film industry, and knew this even when I was your age. I live in north london so emailed EVERY tv/film production company I could find and the ones which replied all said i was too young, which left me feeling so disheartened.
In the end I managed to get a response from velvet TV which is based in manchester (luckily I have family there so was able to stay there for the week) and had the most amazing week which I was able to write about in my personal statement as even though it was 4 years ago it is still relevant to my uni course, so I don't agree with people saying it doesn't matter.
I would recommend trying smaller, more local companies. Many big companies have insurance that means they can't support under 18's, but the smaller ones don't have this. Some ideas are:
-your local town newspaper
-your local radio station
-hospital radio station
-any local based companies which might have a media/PR department
I know this many not seem as exciting as spending a week at a big name magazine or TV studio, but the skills you will learn at smaller companies will be exactly the same and it's still a great place to start.
Wishing you the best of luck, keep persevering! The knockbacks you're experiencing are a massive part of working within the media industry so don't take it to heart too much. And remember that once you're 18, with hard work and determination you will be able to get any amount of work experience/unpaid vacancies to boost your CV so if you end up not being able to get something relevant now, it's not the be all and end all.