In real life, we can have a pencil of length 2 cm. Can we have pencil of length $\sqrt{2}$ cm?
My answer to that was no , we cannot even make 2 cm pencil.
My argument was that when are working theoretically in mathematics we have axioms and certain assumptions. In real life we approximate things, therefore every lengths and measurements are just an approximation.
It is philosophical in nature. Here is my intuitive opinionated answer.
(1) We can easily define a certain Pencil to have certain length. When we say it is $4$ cm then we will get $2$ cm when we break it into 2 . . . .
We can also define it to be $\sqrt{2}$ cm or $\pi$ cm : In that case , it is "harder" to generate $2$ cm.
(2) A Metal bar of size $1$ cm can be heated or cooled to get $1.1$ cm or $0.9$ cm.
Without quantization , we can assume that the Bar goes through all lengths , rational & irrational between original length & final length.
With that view , $\sqrt{2}$ cm or $\pi$ cm will be nothing special . . . .
With quantization , we can take the least change to be $\delta$ : that value will dictate what lengths we can generate , whether irrational or not.
(3) When heating & cooling curves & angles & curved angles , we might generate more irrational values.
Using 3D Volumes like blowing up balloons , we might generate $\sqrt[3]{2}$ & such . . . .
(4) Eventually , it all rests on what values we Define , what values we have , what "operations" we can use & what values Physics allows us to generate.
Electro-Magnetism / gravity / space-time curvature at cosmic levels / macro levels / quantum levels will allow various values while preventing other values.
Yes. We can construct a square with a unit edge length, then the diagonal of that square (the hypotenuse of each isosceles right triangle the diagonal splits the square into) has a length of $\sqrt{2}$. So it's a constructible length which could be manufactured if someone really wanted to. Even taking into account imprecision in measurements, if enough machines were manufacturing enough pencils, one of them could have a length of $\sqrt{2}$ by chance, so it is possible, just not probable.