How irrational quantities physically exist in nature? We know that an irrational no has well defined decimal values upto infinite decimal places. These irrational quantities exist in nature in some kind of measurements. For an example, circumference of a circle is '2πr' , so if radius is rational then circumference will be irrational ,and this case is quite natural. 
But I am unable to understand , how can a physical measurement exist with infinite precision (due to precision of infinite decimal values in an irrational no) ?? 
Isn't it in contrary with the laws of Physics??
I've got bad news for you: Even rational quantities don't exist in nature. Numbers, of whatever type, are just a way to describe our observations. But when we measure, we always have a measurement error, so we cannot really say "this measured quantity has the value $x$", all we can say is "this measured quantity is close to the value $x$". Well, we can do a bit more: Namely quantify how close it is. This is why measurement results are normally written in the form $x\pm y$. For example, a length might be measured to be $1.3\pm 0.5~\rm mm$.

Now we pretend that there is a "true value" of the measured quantity. For example, in the above example, we assume that there is a true value for the length, which is likely(!) less than $0.5~\rm mm$ away from the value $1.3~\rm mm$.

Exact numbers are therefore nothing we can measure, and nothing which we can even prove really exist in nature. They are abstract concepts we made up in our mind in order to describe what we observe. And they work very well for this purpose. But whether they really correspond to something "out there", nobody can really know.

But wait, you say, there are circles, and for those we can prove the circumference over diameter is $\pi$. So $\pi$ is part of our world, right? Wrong: You never have seen a circle. You've certainly seen lots of shapes that closely fit our concept of a circle. But likely if you look close enough, you'll find that it is not really a circle. And even if you cannot detect the difference, how do you know that if you had looked just a little bit closer, you'd have found a deviation?

And ultimately, those circle-lookalikes you've seen probably were all made of atoms anyway, which means they could not have been a circle, as they have volume, while the circle is an infinitely thin line.
Certain natural phenomena in probability theory such as Buffon's needle experiment have answers containing an irrational constant, namely $\pi$.  Therefore such constants are arguably "in nature" (or any reasonable idealisation thereof) even if you are not particularly interested in the circle itself. The fact that there is an exact formula for the probability that involves an irrational number is precisely an argument against limiting precision in this fashion. Mathematicians have great respect for quantum theory but that theory involves a different type of idealisation of nature.