You can say: having lunch, taking or having a lunch break, eating lunch, ordering takeaway lunch, ordering in pizza for lunch, going home to eat lunch, going out for lunch, having lunch in a cafe.

By themselves, they sound strange. So "having lunch" or "to have lunch" would be better. "He is taking his lunch elsewhere" - means that he took his lunch (the food) to some other place before eating. "He is eating his lunch elsewhere" - means that he found another place where he can have eat his food.

If I'm not mistaken, neither of them is correct. I would say "to have lunch"... But I'm not a native speaker..

I think eating lunch is correct, and having lunch is also correct.

1. eating lunch 2. taking lunch 3. having lunch All three are fine.