Does an interrupt handler have to be reentrant?

Does an interrupt handler have to be reentrant?

I'm using a static variable inside an interrupt handler, making the
interrupt handler non-reentrant.
Is it OK to have a non-reentrant interrupt handler?
When a hardware interrupt is raised, does the event go in some sort of a
queue and wait for the current interrupt handler call to finish or does
the interrupt handler get called right away?
Thanks
PS. I'm using C if it makes a difference.