Suppose I am in the center of a hypothetical 4mx4m room and there is no gravity. I am wearing a spacesuit with no means of propulsion (e.g. venting, ejecting, etc..).
Would I be able to reach "the door", i.e. start movement in a certain direction, by body movements of some kind, or would I float in the center of the room forever?
You haven't specified if you're in vacuum or not (though the spacesuit is suggestive).
If you're in vacuum, you're stuck, unless you started with some motion relative to the room, or something accelerates the room itself.
If you're in a fluid medium, even air, you can "swim", pushing the medium around you preferentially in one direction to accelerate yourself in the opposite direction. This would be slow going in air -- each stroke produces very little momentum -- but you could escape.
You haven't mentioned your initial velocity in respect to the door. It is probably not perfectly zero, so after some time you will be able to touch a wall, floor or ceilling. Then you can push yourself towards the door.
I would say, the chances your initial velocity is really 0 is almost non existing.
You cannot change the location of your center of mass without some form of propulsion. You can, however change your attitude by wiggling around like a falling cat.
