If it's encrypted, how would the police read it? The secure place would be a safe.

I imagine if there's some standard out there where you can use the public key to encrypt it and law enforcement, the medical community and other authorized users have access to the private key this could happen (though that sounds like a recipe for the private key to be leaked) ... but this system doesn't exist yet.

Even a small thumb drive has enough space to hold your entire medical history -- even the 700 pictures of you taken in that CT last year would only take a little bit of it. But with your name and date of birth and primary physician any doctors can get at your entire medical history too, and there's little (additional) risk of it being lost or not noticed for what it is.

You could put your entire medical history on the drive (unencrypted!) and make a card like I suggested that says go to the drive for more information ... but then if it's lost or stolen, you might think it's a problem. And the doctors may not be able to immediately read your flash drive for the same reasons I gave that the police might not be able to immediately read it.

Ultimately, unless your medical issues are really unusual and complicated, it seems to me you could easily put everything a doctor would need to know on a low tech credit card sized piece of paper. The only tech needed would be a printer to make it as small as possible as needed to fit everything you want on it, and the only tech needed to read it might be a magnifying glass if you went too far.