Note: This is not a work of scholarship so I have included no references, but all of the claims that follow are true – as can easily be verified by using google searches on the terms used to find links to authoritative documentation.
A random variable which is binary has just two distinct possible values (such as a tossed coin of negligible thickness always landing Heads or Tails – with the edge-based landing being of essentially zero probability, and also quite unstable even when it occurs).
Many people allege that human sex is binary because most humans have two sex chromosomes, one of which is always of ‘X’ type and the other may be either ‘X’ or ‘Y’. But these are not the only possibilities. There are also humans with XXY XYY and various other combinations, which while “abnormal” are vastly more common than edge-on coin landings. So even just in terms of chromosomes, sex is not binary.
In fact, although the number of possible chromosome combinations is finite, the actual distribution of sexual characteristics is not even discrete. This is because what we identify as the sex of an individual is not determined by the chromosomes alone. What might be expected from the genome in a normal environment can be changed substantially by external factors such as hormones (or hormone-emulating pollutants) absorbed from the mother’s blood, and/or by the effects of other genes such as that for androgen insensitivity syndrome (which can produce a completely normal-looking female form from an XY zygote in a perfectly normal external environment). Sometimes such an individual lives a full life as an apparently very feminine (albeit usually infertile) woman, and other times an infant identified at birth as female develops various kinds of male characteristics at puberty – or conversely, a male-appearing infant may develop various kinds of feminine characteristics (either physical or just mental).
So far as I am aware there is really no sexual characteristic that does not exhibit an essentially continuous range of values; and although sex could be defined as a binary property by choosing an arbitrary dividing point for some particular variable (such as relative location of urethra and erectile tissue), it is possible that the partition would be different if based on different variables (such as the effectiveness or location of testes).
So the claim that “science says that everyone falls into just one of two clearly defined sexes” is entirely wrong.
However it is also true that there are various characteristics commonly used to define sex that are strongly bimodal. This means that the variable in question has a possibly continuous range of possible values with two identifiable peaks in its probability distribution (such as the distribution of body mass of black widow spiders – but not the combined distribution of heights of men and women because the difference in mean heights of men and women is too small relative to their standard deviations to produce bimodality when the two distribution curves are combined). Of course there are sex-related properties of humans that are more definitely bimodal with a substantial separation of peaks. (Examples include amount of erectile tissue, relative position of urethra, location and functionality of gonads, etc.) BUT there is still an apparently continuous range of intermediate forms, and it is possible for an individual to be in the “male” range with respect to one such variable but “female” with respect to another.
See also: Spurious Appeals to “Science” | alQpr