Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? In this question  "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\mathbb{R}^2$ as an application of basic field theory. After appropriate definitions of the possible construction steps, 
four of these, namely Hornfeck, Jacobson, Lorenzen, and Meyberg, prove that $\sqrt[3]{2}$ is inconstructible starting from $\{0,1\}$ or $\{(0,0),(1,0)\}$, respectively, but then conclude without further justification that the duplication of the cube is impossible.
For a while I believed this last step to be obvious. But now, having to teach this for the first time the day after tomorrow, I have doubts: Being given a cube, say in $\mathbb{R}^3$, should mean  being given its eight corners, and then I could use these to do constructions in space, using lines through two points  and circles around one point and through two points. Or, to put it differently, I could take any three noncollinear points given or already constructed and do plane constructions in the plane spanned by these.  Restricting the constructions initially to one particular coordinate plane containing one face of the cube appears unjustified to me. 
My specific questions are:
How does one treat this problem honestly and elegantly, with a minimum of coordinate computations?
Is the problem I see perhaps the reason why the fifth of my books, by M. Artin,
does not mention cube doubling?
 A: I think it is straightforward to prove by induction that starting from the eight vertices $(\pm 1, \pm 1, \pm 1)$ of a cube in $\mathbb{R}^3$, the constructible points in the OP's extended sense all lie in $K^3$, where $K$ is the union of all Galois extensions of $2$-power degree over $\mathbb{Q}$. It follows that the possible distances determined by the so constructed points also lie in $K$, whence $\sqrt[3]{2}$ is not among them.
In short, the cube cannot be doubled even in the OP's extended sense. On the other hand, it will be hard to verify what the oracle of Delos had in mind.
A: Disclaimer: The following perhaps isn't an answer to your question as stated, so my apologies if this answer is useless to you. However, you're asking for how to treat this problem "honestly", and I think that adding the right kind of historical perspective falls under the heading of honesty.
Anyway, I think it is important to observe here that the ancient Greeks themselves did not limit their solutions to plane constructions. As can be read in Sir Thomas L. Heath's A History of Greek Mathematics, Vol. 1, pp. 246-9, Archytas proposed a solution to the problem where he intersected three surfaces of revolution in Euclidean $3$-space (a cone, a cylinder, and a torus) to obtain a point whose coordinates generate the field extension $\mathbb{Q}(\sqrt[3]{2})$. 
It is somewhat misleading, I think, to keep referring to these problems as "the three famous unsolved problems of Greek mathematics", because the Greeks in fact solved them many times over, Archytas' solution being only one out of a multitude. Moreover, they even recognized that the solution could not be achieved by plane methods, in a way: Pappus has it that the Greeks classified construction problems as "plane", "solid", and "linear", according to the methods with which the problem could be solved. Of course, they never tried to make this very precise, let alone tried to prove it, but then they weren't trying to do the impossible either.
A: You can define the problem however you want in your classroom. If you think that three dimensional operations should be used, make a list of which three dimensional operations you think are allowed and work out which field extensions they will give rise to. If there are all quadratic, then the result is still true.
The one time I taught this, I decided modern students had no reason to care about straight edge and compass, and just talked about a pocket calculator with $+$, $-$, $\times$, $\div$ and $\sqrt{\ }$ keys. (Of course, soon, no one will care about this either.)
