One of the ways to get into this is through your school/college. Some have links or agreements with NHS trusts to allow placements, and there are sometimes programmes in place already. I'd recommend speaking to your careers advisor at your school to find out if anything already is in place. Also it can be good to look outside hospitals, places like hospices, clinics or health centres are a really good experience, or even nursing homes (although you aren't necessarily getting to work with doctors, you can still learn a lot about complex health needs). Any experience where you are working with vulnerable people is a useful one and as a doctor you're going to need to communicate with people from all walks of life.
You should also consider how you are contacting these places and the way you're asking for work experience too (are you showing your passion, explaining why it would be useful to you, explaining what you want from your career), and remembering that it's difficult for employers to meet demands for specific times and lengths of placements. Even shadowing for a day or two is better than nothing. Making sure you follow up any contact is important as you should be getting a response either way.