What is the length of a point on the real number line? Since an interval is made up of an infinite number of points, I am considering the relation of the length of an interval and the length of a point, this lead me to ask what is the length of a point on the real number line ?

The nested interval theorem made me feel the length of a point should be $0$ because of $\lim_{{n\to\infty}}(b_{n}\!-\!a_{n})=0$

While if the length of a point  on the real number line is $0$, then I get a contradiction : Supposing we remove the point $0$ on the real number line, then there is a gap there, and the width of the gap is $0$ since the length of a point  on the real number line is $0$, however I think the width of the gap being $0$ is equivalent to there being no such gap on the real number line, so this leads to a contradiction. 

What's wrong here? Does a point on the real number line have a width? If so, what is the length of a point on the real number line? Infinitesimal?
Much like many other 'paradoxes' you'll find that carefully formalizing the problem makes it go away. Whatever counter intuitive phenomena persist is not paradoxical, but simply counter intuitive. 

So, what do you mean by length of a point? Well, whatever it is it seems reasonable to define it to have the following properties: 1) the length of an interval $[a,b]$ is $b-a$; and 2) if the point $p$ is contained in an interval, than the length of the point is $\le $ the length of the interval. We further assume all lengths are measured by non-negative real numbers, so there are no infinitesimals around at all. 

Now, from these it follows that the length of any point is $0$. So, any notion of length of a point conforming to the above must assign length $0$ to each point. That is a theorem. 

Consequently, the situation you describe, where removing a point having length $0$ results in a broken line is a correct description. It is not a paradox but rather a counter intuitive situation. Well, what shall we do with it? You can examine our assumptions and change them, or hone your intuition. In this case, I suggest the latter.
Yes your are absolutely correct.I argue that length of a point is infinitesimal,so is breadth of a line segment.An interval(a,b) is a disjoint union of points.If point has a length 0, 0+0+.......infinity =b-a-finite is impossible.So it should be infinitesimal.Same argument holds for breadth of a line segment.A rectangle of sides a and b whose area is ab is formed by placing a line segment of length a arbitrarily close to one another.If the line segment has breadth 0 then 0+0+......=ab-finite is impossible. similarly we can prove that thickness of a plane rectangular area is infinitesimal. My final conclusion is that length,breadth, thickness of a point is infinitesimal.point has no length is wrong.it is arbitrarily small  . I think this proof clarifies your doubt