Matt Yglesias asks an interesting question: “If Elves, Hobbits, and Men can interbreed in Middle-Earth, aren’t they the same species?”
I’m not sure if the ability to breed entirely defines a species (as Yglesias mentions, we interbred with Neanderthals, and I still think they’re classified separately, though some say otherwise), but I’d call it a strong indicator. So let’s take a look at the interbreeding possibilities in Tolkein’s universe, one-by-one.
Men and Hobbits. I seem to recall Tolkein briefly mentioning in one of the Return of the King appendices that Hobbits were originally “of man kind”‘ as were the Stoors (Gollum’s species) from whom Hobbits may have descended. So we would classify Hobbits as a race, or at most a sub-species, of Men.
Men and Elves. There are but few examples of human-elf unions, but they involve the most significant characters I. middle-Earth. When they do happen, the offspring are referred to in the Silmarillion as Half-Elven, and their genetic inheritance seems to be a throw of the dice. Two of these Half-Elven, the brothers Elrond and Elros, were actually given the choice of what kind of being to live as. Elros chose Men, became the first King of Numenor, an is thus the ancestor of Elindel, Isildur, and through many generations, Aragorn. Elrond preferred to be an Elf, lived through toe Ages of Middle-Earth, and fathered Arwen. Thus the marriage of Arwen and Aragorn unifies the two lines of Half-Elven. When Aragorn dies, Arwen, though at least a nominal Elf, lost her immortality and suffered a lonely and mysterious death. Their son, Eldarion, appears to have no choice in what race to live as.
What does all this mean? To me it means that elves are not Men, and men not Elves, but the status of Elf can be lost and transferred by non-biological means, which is to say, divine grant and judgement. Thus, Elves are a kind of Divinely-Ordained Super-Men, distinct but related. The Silmarillion refers to both as the Children of Illuvatar (the God of Middle-Earth), and they have other titles suggesting status as sibling species. Like horses and donkeys, except their offspring are not sterile.
Men and Orcs. The wicked race of Middle-Earth are the tortured and mutilated offspring of elves captured by Morgoth (Tolkein’s Lucifer, the forerunner of Sauron) in the First Age. So I would assume a similar dynamic at work.
Men and Dwarves. Dwarves are unique, Neither Children of Illuvatar nor perversions of Morgoth, Dwarves were carved out of stone by one of Illuvatar’s demiurges, who breathed life into them. Illuvatar suffered them to live when they demonstrated sentience. I do not recall any evidence of any interbreeding between these most insular creatures and any other species.
Other Combinations. If Hobbits are Mannish, then they would be able to produce very short Half-Elfs and Half-Orcs if the situation required. Elves should likewise be able to breed with Orcs, and offspring might be at least Half-Elven.
All of which doesn’t explain why Aragorn couldn’t just get together with Eowyn. Unless it does.