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You can do it by first constructing a right triangle with sides of length 1, 1.
The hypotenuse is length square root 2. This is easy to do with a compass and straight edge. Now you can use the compass to make a new triangle with one side square root of 2 and the other length 1. The hypotenuse is on length square root 3.
Keep doing this and you will get an hypotenuse of side square root of 13 exactly.
It is a pretty straight forward geometric construction.
( Hint to start, take a line you call length one and construct a perpendicular to it with compass and straight edge. Make the other side of the triangle length 1 also. Hypotenuse is length square root 2)
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