Even Greener Pastures

You probably know this already, Nikola, but 'be sick' only means 'throw up' in BrE.

"I was sick yesterday" (AmE)  = Stative verb, meaning "I wasn't well", in general terms
"I was sick yesterday" (BrE)    = Active verb, meaning "I threw up"  ( but may otherwise have been fine - only refers to a single incident).

Here's another one for the list... 'chunder'.

It can also mean to throw up in AmE. So as you say, it can mean a general unwell feeling or to vomit in AmE. I've heard people say "He got sick all over the car." I'd say though that in general, we say throw up, vomit, etc.

But to get back to the question, I'll reiterate that you can forget about the whole taken sick, taken ill, etc. I'll agree with Neal that it's archaic in AmE. I would guess that any American under 50 would give you a funny look if you used it.