How can points that have length zero result in a line segment with finite length?
This may seem rather a strange thing to say, but I don't think it's helpful to think of lines as made up of points: the "lininess" of a line is an inherent property that points don't have, so it has some extra qualities that points don't, such as length.
The real numbers are basically the answer to the question "How can I augment the set of rational numbers so that I don't have to worry about whether limits that ought to exist really do exist?", from which one can then do calculus. One can wheel out $\sqrt{2}$, $\pi$ and so on if one so desires as an obvious example of a point where one needs this.
Perhaps a more helpful introduction of the real numbers is to say "I want to know how far I am along this line." You then say "Am I halfway?" "Am I a quarter of the way?" "Am I 3/8ths of the way?", and so on. This gives you a way of producing binary expansions using closed intervals, and you can then introduce the idea of asking infinitely many of these questions (which will obviously be necessary, since $1/3$ has an infinite binary expansion), and the object in which the infinite intersection of the decreasing family of closed intervals with rational endpoints constructed by answering the sequence of questions contains precisely one point is called the real numbers. Hence one ends up with the real numbers as describing locations on the line, while not actually being the line itself.
In fact, the construction of the real numbers also gives you some "lininess" as baggage from the construction: you produce a topology, which tells you about locations being close to one another. This gives the real numbers more "substance" than just being ordered and containing the rationals. One can define topologies on the rationals, but the real numbers' completeness in their topological construction is the key. Completeness forces there to be "too many" real numbers to be covered by arbitrarily small sets. (Obviously countable is too small since the rationals don't work, but the Cantor set shows that one can produce uncountable sets with zero "length".)
One large hole in this so far is what "length" actually is. To do things this way, one is forced to introduce a definition of the length of a rational interval $[p,q]$, which must of course be $q-p$. Since one is not concerned at that point about the interval actually being "full" of points, one can simply introduce this as an axiom of the theory: all of us at some point have owned a ruler and know how they work with integers and small fractions, and it's not too much of a stretch to stipulate that one can have a ruler with as small a rational subdivision as required, without having to resort to infinite subdivision. (Which is another point worth emphasising: without infinite processes, there is no need for the real numbers in toto: one can simply introduce "enough" rationals for the precision one requires, and work modulo this "smallest length".)
This way, one starts with "length" and ends up with "real numbers", rather than trying to go the other way, which is theoretically difficult and mentally taxing and counterintuitive (besides all the Cantorian stuff).
At that level, you can only define length in terms of line segments, which should not present a problem.
To approach the idea of "length" of a point you could use the idea of probability.
You might use an example such as this. Suppose we randomly choose a number $x$ in the open interval $(0,1)$.
- What is the probability that $x$ will be in the interval $\left(\frac{1}{2},1\right)$?
- The interval $\left(\frac{1}{3},\frac{2}{3}\right)$?
- What is the probability that $x$ will exactly equal $\sqrt[3]{\frac{3}{\pi}}$?
You can then relate the idea of "length" to probability.
You write: "As an undergraduate I assumed it was due to having uncountably many points. When I learned about measure theory, I realized that's not the explanation."
But in fact this is indeed the explanation. Lebesgue measure $\lambda$ is countably additive (in ZFC) so if $\mathbb R$ were countable one would indeed have $\lambda(\mathbb R)=0$. But countable additivity is not generalizable to any hypothetical notion of uncountable additivity. Therefore no paradox of the sort $\lambda(\mathbb R)=0$ arises.