Is it possible to carpet my whole room without installing a permanent carpet? Or without removing the floor?

Sounds like you would do well with a large area rug.  You can get a room size rug in your specific size at a carpet store who will bind the edges. Either that or carpet squares to create your own size and pattern.

sure.  Though getting one that big might be pricey...say compared to wall to wall installed carpets.  You got to price them out. I don't see how big a room you got and don't know the prices of carpet in your area.  Carpet is never permanent though it is laid to be permanent.  Mine has been in my house over 40 years. You just don't wear your boots when you been shoveling schit in the pig pens.  Leave the boots outside and the carpet stays cleaner Longer. Common sense.

POSSIBILITY ( noun ) a thing that might happen, the state or fact of being likely a thing or action, that maybe CHOSEN or done out of several possibilities

.... possible is demonstrable and testable   rooms sizes vary as do  edged carpet  meaning matching a edged carpet to ones room size

example 10 x 12 square room get a 10 x 12  or smaller  edged carpet

or at  Lowes  home depot  or carpet stores they have bolts of carpet on roll cut to size and extended sizes  match with carpet tape

you need a floor in a room to put the carpet on  perhaps one mean the existing FLOOR COVERING  such as carpet hard wood   tile  most cases if not  using an adhesive or carpet   tacks    carpet can go over other floor coverings

It depends what the bare flooring looks like.  If the room is currently carpeted and you're thinking of tearing it up and putting down an area rug it may not be that simple.  Unless there is laminate flooring or presentable hardwood flooring under the current carpeting, you'll likely need to have carpet or tiles or some other flooring permanently installed before throwing down any rugs.

Sure it's possible. That's what carpet remnants are for. You simply lay them down on the floor--use a bit of double-faced carpet tape to anchor the edges. 

measure the exact room size and go to a carpet place and buy it at that size and have them bind the edges so it does not unravel.

I have purchased carpet of a size sufficient to cover a room. Had the edges bound and then just put it down like a large rug. 

Yes, it's possible.

An area rug is going to be much more expensive than buying carpet by the square yard from a carpet place. Measure and buy a piece that will cover the whole floor. You don't need to nail it down. 

Area rug, not permanent and sits on top of floor