Yes a rabbit can live happily in a bedroom size area. I know several devoted rabbit parents that have bunnies living in bedrooms. If the room is carpeted they can get traction to run REALLY! fast, they will have room to jump and stretch, they will (usually) jump up on the bed. Your bedroom bunny may even try to make your bed.
Except for special reasons (which I will discuss later in this answer) your bedroom bunny can have full access to the entire room all the time. A baby gate can be used to keep the bunny in the room while the bedroom door is open. Notice that this gate has opening that a bunny might squeeze through, so we used zip ties and plastic lattice to keep her from trying to squeeze through. Also this is very tall gate, it is higher than the door handle so in this configuration it is not possible to close the bedroom door all the way.
You will want to train your bunny not to chew on furniture before allowing unsupervised access to the bedroom. Notice that these pictures show a wood bedroom set without chew marks and Ruby has full unsupervised access to the room 24/7. You will want to block off access to anything your rabbit can get under that you can't reach under, this may include some or all of the space under the bed.
In our case the dresser had an opening at the bottom that Ruby could get under. We pulled out the bottom drawer, cut some cardboard and stapled it on from the inside. This does not damage the exterior of the dresser and it has been effective (no picture currently of this solution). Also note that in the first under the bed picture we placed a thin (1/4 inch) sheet of paneling down between the bed support and the box springs. Some rabbits will burrow up into box springs, others won't. Ruby did not before we did this, but we added it while building the box. The box under the bed is so she can not get directly in the center of under the bed. Ruby suddenly got very sick and she hid under the bed. Getting her out was problematic so we added this box as she began to recover.
The question Can house rabbits have free access to the whole house all the time? has many points that transfer to this question so I won't go over them in detail but a safe place for the bunny to rest is important. Some place that is their own and can be closed as needed. We use a common habitat available at most pet stores, it is placed on the floor so she can get in and out easily. The door is converted from the standard ramp to hinge at the side (zip ties for hinges). Some rabbits like to Jump on top of things, to keep her from injuring a foot slipping through the wire top, we added a board (left over from under bed) and a throw rug for comfort.
Unless we are moving furniture or she is very sick the door is always open. Her litter box, hay rack, salt lick and water bowl are in her safe place and available to her all the time. She gets two meals of pellets per day and those are placed in her safe place, with very little training (a couple of days) she will go on command to her safe place. This makes it easy to get her if there should be an emergency (like a fire).
Rabbits can be messy with hay, we find putting the hay rack on the outside so they can only get hay by standing in the litter box and pulling through a couple pieces at a time a great solution. It is important that the hay rack have a lid you can close, if you don't want them to access hay from the top of the rack.
