Yes... You have to have somewhere to vent that heat though...

Thats why people are going into the attic and such. If you have it exhausting into the room it will get too hot. If you could vent it outside of a window would be fine.

5ft vertical is a bit short for that setup. Is it 5 feet from the bottom of your hood or the top of the closet?

I am in the same type of situation. I needed to vent somewhere but attic was not an option so I am venting out the window. Even then its still hot in the room from the light... Cannot skimp when it comes to ventilation and HID lamps.