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.