Just go to home Depot and rent a sprayer - just need to do a little bit more of preparing the areas (i.e. taping and covering what you are not painting)

I live in an existing house and many of the professional quotes I received involved the painters simply spraying the entire closets (ceilings and walls) with a full interior repaint. Not exactly sure how they would go about it though.

I actually ended up doing it myself and leaving the closets alone, so I am not sure how it would have turned out though.

At the end of the day, they are just closets and will most likely be covered with clothes and junk, so just slapping some paint on with a roller and brush may be the most cost effective way.