Of course $e$ cannot be constructed with straightedge and compass. If we allow a marked ruler or a non-rectractable compass (or we use origami...) we can construct numbers like $\sqrt[3]{2}$, but no trascendental numbers. However the quadratrix of Hippias allows us to construct $\pi$, albeit using a special instrument.
I wonder if a similar instrument was ever devised to construct (in a finite number of steps) $e$.
Take two set squares without marks. Draw two marks on the edge of one of them and paste the ends of a moderately long, inextensible and with uniform mass distribution string over the marks.
Bisect the segment of the set square joining the two marks and mark that midpoint on the edge.
On a vertical $xy$-plane, use the other set square to transport the strung one parallel to the horizontal $x$-axis with the midpoint marked on the previous step over the $y$-axis. It will happen that the lowest point of the now hanging string will be over the $y$-axis and moving up or down this set square you can put it at $(0,1)$.
The equation describing the points of the string is $y=\cosh(x)$. Draw a vertical line passing through $(1,0)$. Mark the intersection of that line and the string at $(1,b)$.
It happens that $e = b + \sqrt{b^2 - 1}$ and you can construct this number in the usual way because you have already marked $(1,b)$.