For gears we use the radius of the gears instead of the arm lengths of lever. We can set up some gears to replicate a lever by having two gears that have a third idler gear in the middle, as in the diagram below:  

[![enter image description here][1]][1] 

The size of the middle gear is not important, as it is only there to reverse the direction of the third gear. If we attach weights to the perimeters as shown in the diagram, Then a mass of M attached to the larger gear with radius 2R will balance 2M attached to the gear with radius R, just like a lever with one arm twice as long as the other.

The torque applied by the weight on the larger gear on the left is $T_1 = M \times 2R$ and the torque on the smaller gear on the right where $T_2 = 2M \times R$ so the torques are equal in magnitude, just as in the case of the lever. 

> "Now the short arm moves 1/4 the distance, but we get 4x the force"

This description in the video is a bit confusing. They are not talking about the distance the tip of the short transfer arm in the middle moves, because that tip moves the same distance as the tip of the long transfer arm in the middle, because the tips are in contact. They are talking about the distance the tip of the short outer arm on the far left moves, relative the distance the tip of the long outer arm on the far right moves. The diagram below is a schematic of the situation at timestamp 1:46 in the video:

[![enter image description here][2]][2]

It is now clear the distance they are talking about is the vertical distance the tips of the outer arms move. 

An even better replication of a fulcrum lever using gears (or pulleys) is to have both gears on the same shaft as depicted below:

[![enter image description here][3]][3]

Now the angle the input lever moves through is the same as the angle the output lever moves through,  exactly as in the classic fulcrum lever case.  


  [1]: https://i.sstatic.net/QHCD1qnZ.jpg
  [2]: https://i.sstatic.net/Z3qsebmS.jpg
  [3]: https://i.sstatic.net/LuTQ86dr.jpg