Is "I turned on and off the computer" grammatically correct?

FumbleFingers' comment points to what's going on here:

... you can't turn on and off a computer, whereas you can step on and off the scales (or walk up and down the road, etc., etc.)

In step on and off the scales both on and off act as prepositions, taking scales as their respective objects, and conjoining the two prepositions implies an ellipsis of two conjoined preposition phrases: on the scales and off the scales. This is consonant with how we express the result of each action:

I step on the scales. > I am now on the scales.
I step off the scales. > I am now off the scales.

But in turned on (or off) the computer*, on (or off) does not act as a preposition but as an adjective designating a state which you cause the computer to assume. Note that with turn, unlike with step, on (or off) can be posed after the object:

I turned on the computer = I turned the computer on > The computer is now on.

The preposition phrases implicit in the conjunct preposed prepositions are not valid. Consequently we must postpose the conjunction:

I turned the computer on and off.