Is heights of humans actually a discrete random variable? Suppose the human population consisted of $N = 3$ people, each with a specific height. Let $X^N$ be the random variable representing the heights of this population of $N$ people. Since $X^N$ can only take $N = 3$ distinct values it is a discrete random variable with a probability mass function.
For example, we could have $N = 3$ people with heights $150$ cm, $160$ cm, and $170$ cm, and thus the probability of any particular height occuring is $1/3$.
Now, consider the case of $X^N$ when $N = 6$ billion, i.e. the heights of the real-world human population. We now have $6$ billion distinct values for $X^N$. Although there is now a very large range of values $X^N$ can take, it is still a discrete random variable as those $6$ billion discrete values are the only values of $X^N$ that can occur.
Therefore, heights of humans is actually a discrete random variable and not a continuous random variable? Everywhere I look it says that human heights is a continuous random variable with a pdf, but it seems from the above that it is actually a discrete random variable with a pmf?
 A: 
.. each with a specific height that can be measured with infinite
accuracy..

Based on this, we could say that height of a single individual, say $X$, is continuous RV, it can be any real number within a plausible range. This makes vector of heights, say $X^N$, a continuous random vector as well.

For example, we could have =3 people with heights 150 cm, 160 cm,
and 170 cm, and thus the probability of any particular height occuring
is 1/3.

This is only one realisation of the random vector described above. The probability $1/3$ describes something like
$$P(X_1=150|\text{Three people have heights 150,160,170})=1/3$$
which is not equal to $P(X_1=150)=0$. (because a continuous RV being equal to a specific value is $0$)
If $X$ was measured with finite precision, it'd be discrete RV in nature, and even a very large $N$ like 6 billion wouldn't change the nature of $X^N$, which would be discrete.
A: Height is continuous in principle, but reported as one of various discrete measurements in practice.
What's more, there can be many minutely different conventions both within and between datasets.
For example, in some countries original measurements might be variously in inches or cm (mm) and standardized to one or the other. That can lead to a distribution that is oddly spiky in detail, although it can take a very large sample to make that obvious.
Even if a country or a group of researchers uses just one of those units of measurement, the detail can still be complicated. In practice, observers can use different personal or shared rounding rules and there can be digit preferences (e.g. a tendency to report heights ending in 0 or 5 mm rather than any nearby values).
Although there is plenty of scope to make this rigorous by introducing notation and talk of different sample or outcome spaces, for many readers all that may be needed is a careful contrast between principle and practice.
To the fair comment that height is just an example here: sure, but there is a similar story about many other variables.
A: A random variable $X$ is defined in terms of possible outcomes $\Omega$, not observed outcomes from trials. If $\Omega$ is defined to be countable, then $X$ is discrete, and if $\Omega$ is defined to be uncountable, then $X$ is continuous. Any finite set of draws of $X$ will not only be countable but finite, whether $\Omega$ is uncountably infinite, countably infinite, or finite.
In other words, we choose $\Omega$ to represent the possible outcomes as a modeling decision about what the possible values are. It may be useful to decide to model height as a continuous random variable because this decision allows us to use well-understood probability distributions, to have well-grounded notions of distance and ordering, and to estimate models that fit the data well. Even if this decision is wrong for fundamental physical reasons related to limits on what distances can be measured physically (N.B. I am not a physicist), it may be more useful than a discrete model with billions of ordinal outcomes.
