Irrational numbers in reality I have a square tile which measures 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another will be $\sqrt 2$ metres. However  $\sqrt 2$ is an irrational number, could someone explain how it is possible for a non-terminating and non repeating number to represent a finite length in reality? 

If I was going to get upset about the 'reality' of any real numbers that are irrational, the square root of $2$ would not be on the top of my list.
Put on your "I am Pythagoras with ruler and compass in hand' hat. Using mind experiments you construct all the positive rational numbers on a 'ruled line'. Working in the plane, you construct the length $\sqrt 2$ and mentally place it as an offset from $0$ on the line. You know it is something new, but you can get arbitrarily close approximations with rational numbers.
You don't know about decimal representations and when you are told how they just 'go on and on' you yawn - you've already come to terms with the 'reality' of the situation.
As you look into the nature of real numbers you might learn that there is something called a computable number; the $\sqrt 2$ is a computable number.
Here is an irrational number that would be on my list of 'hard to fathom' irrational numbers.
Keep flipping a coin with outcomes $H = 1$ and $T=0$,
$\quad (a_1, a_2, a_3, \dots)$
and 'build' your real number
$\tag 1 a = \sum_{n=1}^\infty \frac{a_n}{2^n}$ 
Now that is not easy to imagine, but since we don't want 'gaps' in $\mathbb R$ it is included as an idealization.
