Do I have to have done work experience to get a place at medical school?

There are very few things any university will specific you HAVE to actually do, the more important things are showing that you have made an attempt at gaining relevant experience, and even more importantly showing that you have learnt relevant lessons from what you have managed to do.

For example, it can be a nightmare getting work experience in a hospital itself, but there are plenty of places (like care homes) that will be much more willing to take you in. What universities want to see if that you have a basic understanding of health care settings and you have a realistic expectation of a career in that field (and if you do work at a care home you will know it can be particularly grim at times). Secondly, it is important that you don't just say you did it and stop there. A candidate who worked somewhere seemingly irrelevant, like a local tesco, who could describe the teamwork and organisational skills this taught them, could do a lot better than someone who managed to get a really high flying specialist hospital placement, but who just relied on the placement itself to sound impressive and didn't reflect on their experiences there.