Ok, now I'm confused. Assuming by "box" you are referring to a Roku/Apple TV/Fire TV, why can't you do this? Is it the lack of DVR access you are talking about?gimme some examples.. Currently I have DirectTV with 2 DVR's, one regular and one Genie, plus 6 more HD boxes, the whole house can watch anything from either DVR.. I also have 3 Apple TV's one in the LR, Playroom and my Bedroom.. If I could ditch DirecTV and only have one "box" in each room and maybe one in my Network closet I would drop it in a second.. I also have wired ethernet behind every TV.
If you try to use one account per service for the entire house, you may run into some issues. Netflix only allows 2 streams at the same time. Hulu officially states you can connect one account to as many devices as you want but only stream to one device at a time. However, I've read where people have tested this and had 6 streams watching different content from the same account at the same time. No idea on other services.
If you have a large collection of movies then Plex is probably the easiest and most obvious route.
Here, let's do it this way. List the requirements you would have and we can group-think it.