No reason you can’t paint most of the room a dark read. The most important area to have completely black it that nearest the screen and as Michael said black velvet (or another light absorbing cloth I would add) is preferable to flat black paint because if you get the right type it will have less sheen than even the flatest black paint you can make.
This makes painting the room a dark red and curtaining the area around the screen and several feet out a nice option. That’s the route I’ve taken except the curtains go all the way back to the back wall to minimize reflections in my peripheral vision. The only thing painted black is the ceiling and surprisingly (well to me at least) a black ceiling and burgundy walls actually looks nice.
Though painting black will cost less and likely be less hassle than curtains having tested both I would go with low sheen black curtains over paint any day. It’s less reflective. Cuts down more on visual distractions like light swiches. Gives the option of opening them revealing more “friendly” coloured walls should you want to have a sports party or something. And just plain looks more “theater-like.”
How far out to go with curtains depends on how big your screen is in relation to your room and how much you want to minimize visual distractions in your field of vision.