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For a moment I would like to keep Quantum science apart and just dive into Mathematics. It grants us to measure a side with integral length to infinite accuracy (i.e. if a line is of $2$ units then it can be said with full certainty that it has $2.000...$).

Given a right triangle with $base = 1$ unit and $height = 1$ unit, Maths allows us to measure the side with infinite accuracy but just when we join those two lines to get a hypotenuse then we are left with an irrational number $\sqrt2$ about which we are certain that we won't ever be certain (about its length).

My question is, what is the difference between two perpendicular lines and a line which is inclined? What makes it so different than a base (or height) such that it is unmeasurable ?

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    You say you don't want to involve QM (by which I suppose you mean any physical situation where you would actually have to measure something), but your idea of "measuring" something seems very "physical". If Pythagoras tells you that the hypotenuse is $\sqrt{2}$ long, then you *do* know it to infinite precision. I could also say that if you were measuring the lengths with a ruler, how would you know that $1.000\dots$ actually continued to be zeros all the way? As a third option, just define the hypotenuse as the unit for your ruler... then the other two sides are irrational. – 2017-01-28
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    @Lovsovs Got the point. Can you re-post this as an answer ? – 2017-01-28

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There is nothing wrong with any of them. It is just that the equation $x^2=q$ does not have rational solutions for all $q\in\mathbb Q$. This is why real numbers extension had to be introduced.

So there is no common length that would measure as an integer both the hypotenuse and the sides.

If base=3 and altitude=4, the hypotenuse can be measured with infinite accuracy.

It is similar with how equation $x^2=-1$ led to the extension from real numbers to complex numbers. In this extension, any polynomial equation has a complex root, so there is no need to further extend for this purpose.

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Nonsense.

If you allow for exact measurements of the sides of the triangle, that is, if you define the lengths to be exactly $1$ unit, then I am well within my rights to say that the hypotenuse is exactly $\sqrt{2}$ units. $\sqrt{2}$ is well-defined, and there is nothing uncertain about this number.

If we do not define the lengths of the triangle, then measuring the sides to be "exactly $1$ unit" and measuring the hypotenuse to be "exactly $\sqrt{2}$ unit" both equally require infinite accuracy.

In both cases, there is no difference in the measurability of irrational numbers and rational numbers, and irrational numbers are not "less precise" or "unknowable" or anything of the sort. Root-2 is root-2.

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(This was originally a comment, but posted as an answer, as per request from OP)

You say you don't want to involve QM (by which I suppose you mean any physical situation where you would actually have to measure something), but your idea of "measuring" something seems very "physical". If Pythagoras tells you that the hypotenuse is $\sqrt{2}$ long, then you do know it to infinite precision. I could also say that if you were measuring the lengths with a ruler, how would you know that $1.000\dots$ actually continued to be zeros all the way? As a third option, just define the hypotenuse as the unit for your ruler... then the other two sides are irrational.