Is it possible to make a small plane using server fans?

No.
They don't push nearly enough air.
Fan blades are the wrong shape and profile.

The thing in your video does not go nearly fast enough to leave the ground.
Don't forget, you have to add all the control mechanism.

You can buy a small quadrotor that will achieve controllable flight for $50.
 
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Looks like the same ducted fan thing as in the posted video.

I'm sure, the blade design seems a bit different though.

The RC fans could have been made with lighter material and more powerful motor too.

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