Single Cat6 with Multiple Outlets

Is it possible to wire a room with one Cat6 and have multiple outlets for that single line?  Assume I would like a outlet every 6 feet, knowing that only one outlet will ever be used at a single time.
 
I have a picture of what I am talking about, but I cannot post here being that I am a new member...
 
Thanks for you help in advance...
 
 

No, you really can't do that.  What you are describing would be a series of taps along a single cable.  If you were to use one of the middle taps, the far (unused) end would cause additional reflections on the cable, giving you signal quality problems.
 
About the only way to accomplish something close to what you are describing it to put 2 jacks in each wallplate location, and run a cable from one of the jacks to the next upstream  wallplate and another cable from the second jack to the next downstream wallplate.  Then, install a short patch cable between the 2 jacks at each wallplate to form a continuous circuit.  When you want to use a particular wallplate, you would remove the patch cable and plug your device into the jack that connects in the direction of your switch or router.  By removing the patch cable, you've removed the "stub" that would cause reflections. 
 
Having a lot of patch cables and cable terminations will cause some signal degradation, but if you do a good job on the connections and use good quality patch cables, it will be ok.
 
Personally, I would just home run a cable from every wallplate back to wherever your switch or router is located.   Put a patch panel there, if necessary, so you only need to connect the cables that are being used to the switch/router.

@RAL has it right.  Just pull the wires.  
 
As for posting, there are other picture websites good for hosting pictures (photobucket, instagram, etc).
 
Every 6 feet?  That'd be overkill.  They require that for electrical outlets as a means to keep people from using extension cords all over the place.  But where you place network-connected gear is perhaps a lot more predictable than all sorts of other electrical devices.  
 
With some planning you can minimize just how many locations would genuinely need it.  That and be 'close enough' should something change.  Wire is relatively cheap, especially when installed while walls are open.  Even if you don't use it, it's a LOT cheaper/easier than pulling it later.
 
One of my wiring regrets was not moving a media outlet (2 RG-6, 2 CAT5, 1 CAT6) when a window in the guest room got moved (to make for better furniture placement).  Left me with an outlet now under a window instead of tucked behind some furniture.  These being spray-foamed exterior walls means I'm stuck with it.   Hard to keep all the details straight when building a new house...  Live and learn.

If you really wanted this solution, but really didn't want to pull those wires all the way back to the closet, what you could do is pick one spot in the room and run individual runs from there to each of the secondary outlets (it home run them all to one spot in the room) - then also have the "uplink" to the main closet at the same location - then using a network switch there in the room you could multiply the outlets.
 
The only real scenario I'm thinking of here is a bar or lab station where computers would come/go often and need to hook up wherever they sit - but it's an in-between option.  However as said above, that's rarely necessary - if you know you'll only ever need one, then you'd be surprised how easy it is to run the wire around the room if necessary to hide it.  Network cables actually fit pretty easily between the carpet and the baseboard making them disappear completely under the board.  I've even used recessed 45-degree jacks so the wire basically shoots straight down so furniture can be pushed up against flush as well.