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.

What's wrong here ?

This is where you go wrong:

I think the width of the gap is $0$ is equivalent to that there is no such gap exists on the real number line

Much like there are objects with length $0$ (namely, single points), there are real gaps of length zero (namely, the absence of a single point).

As others have pointed out, the length of a single point must be zero for any reasonable definition of length. Notice however, that "length" and "number of points" are very different ways of measuring size. For a point the first measure gives zero and the second one gives one. For an open interval the first measure gives the distance between the endpoints and the second one gives infinity. A single point is negligible in the sense of length but not in the sense of amount.

If you remove a single point from an interval, the total length does not change. The number of points also stays the same, but only because the interval has infinitely many points to begin with.

These different ways of associating sets with sizes are called measures. The measure corresponding to length is called the Lebesgue measure, and the one corresponding to number of points is called the counting measure. These may or may not mean anything to you at this point, but you will encounter them later on if you continue working with real analysis.

I should probably also point out a possible false reasoning, despite being absent in your question: The interval $[0,1]$ consists of points, so its total length must be the sum of the lengths of its points. (We can make sense of uncountable sums. Especially if all numbers are zero, this is easy: then the sum is indeed zero.) But all points have length zero, so the length of $[0,1]$ is zero, too! This is not a bad argument, but it turns out that lengths — or measures in general — do not have such an additivity property. However, if you only take a finite or countable union of disjoint sets, the length of the union is the sum of the lengths. (I'm ignoring technicalities related to measurability as they are beside the point.) The set $[0,1]$ is uncountable, so this "naive geometric reasoning" fails, and it can be enlightening to figure out why.

Take for example the rational numbers. They can be considered as having infinitely many gaps, because the don't include the irrationals. However, those gaps have no "width", because you can come arbitrarily close to each irrational from above and below with a rational number. The situation is similar if you remove a single real number from all real numbers.

In other words: There are just to many of them, so you wont notice if a single one is missing ;).

Strictly speaking the "number line" does not have a "length" automatically associated with it- we have to add that. Typically, we define the length of the interval [a, b] (or (a, b), (a, b], [a, b)) to be b- a. Using that definition of length, the length of a single point is 0.