Hospital

That is what I said. I would have stood very near to it until the next stop, then got out and walked up the stairs (or taken a different lift)

Isn't there an alarm button in a lift?

It's a question of degree, crun surely.
If the strangeness is enough to frighten other people then that implies strangeness of a threatening sort, from which other service users might reasonably expect some protection.
No one has suggested arrest or removal from the premises, have they?

Plenty of people think I'm strange, but I don't fancy getting arrested for it, or bounced out of the hospital.

And don't make eye contact, although be aware at all times.

But if he was genuinely odd and frightened you then best not to show it. I went on a women's self defense course once and the first bit of information we were given was that most attackers are looking for an easy victim.

We were told hold yourself well, walk confidently, keep your car keys in your hand, etc..

Perhaps he just seemed 'strange' to you, but had had bad news, either for himself or a relative.

You don't say in what way the man was strange jenny but I would have just got out of the lift at the next floor.
I was under the impression that lifts have CCTV, though not sure about that.

Make sure you get in a lift with lots of other people or else by yourself.

There IS security in hospitals, it is needed for the nurses, who sometimes are attacked by the very people they are helping.

I am not sure how a security man could patrol lifts. You would need someone permanently stationed in each at all times. There is usually a panic button in a lift, you could use that in an emergency. Someone would come running.