Cuboid computation

You are given a ruler which can be extended indefinitely, and an unlimited supply of 1-inch sticks of negligible diameter. They may be glued together to form a cuboid.

If you construct a cuboid, you can measure the length from one vertex to another, and in doing so "compute" the square root of a certain number by measuring a particular diagonal.

What is the natural density (or asymptotic density) of positive integers whose square roots cannot be "computed" in this way?

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