Here's the official answer from SE's lore team on breeding between the five races.
Tl;dr: It is possible, and does happen. It is rare, however, due to 'cultural differences'.
So what can we draw from that? It seems that while the 5 races coexist amicably in Eorzea due to their mutual dependency and the trials they've endured together, each of the races maintains the uniqueness of their own racial culture enough that pairing with their own race is very strongly preferred.
This is interesting, given how long the races have shared Eorzea by this point. Eorzeans seem to show a greater resistance to the kind of cultural 'bleeding' and blending we witness on earth when disparate groups come to live together over time.
Perhaps Eorzeans are simply a proud lot, who value their racial heritage enough to make efforts to protect it. This can clearly be seen in the Sea Wolf Roegadyn, who still name their children using the old tongue, despite the fact that these names are often difficult for the other races to pronounce. In my head canon, Highlander Hyur (Ala Mhigans in particular) are similarly proud regarding their heritage.
Or perhaps it is partly because there are significant psychological differences between the races that the desire is not often there, or which makes interracial pairings difficult. At this point, it's tricky to say. But SE has stated that we'll see more lore regarding this at some point in the future, so we can look forward to seeing that.
So the long and short of the issue is; given what SE has said regarding interracial breeding in Eorzea, it's reasonable to make a half-breed character.
Also, while Eorzeans are typically a tolerant bunch, given how rare half-breeds are, it's up to you whether your half-breed character would face prejudice regarding their background.
Other considerations include what Roswyn has already mentioned; the significant physical differences between some of the races make certain combinations... Improbable, at least until SE says otherwise.
Tl;dr: It is possible, and does happen. It is rare, however, due to 'cultural differences'.
So what can we draw from that? It seems that while the 5 races coexist amicably in Eorzea due to their mutual dependency and the trials they've endured together, each of the races maintains the uniqueness of their own racial culture enough that pairing with their own race is very strongly preferred.
This is interesting, given how long the races have shared Eorzea by this point. Eorzeans seem to show a greater resistance to the kind of cultural 'bleeding' and blending we witness on earth when disparate groups come to live together over time.
Perhaps Eorzeans are simply a proud lot, who value their racial heritage enough to make efforts to protect it. This can clearly be seen in the Sea Wolf Roegadyn, who still name their children using the old tongue, despite the fact that these names are often difficult for the other races to pronounce. In my head canon, Highlander Hyur (Ala Mhigans in particular) are similarly proud regarding their heritage.
Or perhaps it is partly because there are significant psychological differences between the races that the desire is not often there, or which makes interracial pairings difficult. At this point, it's tricky to say. But SE has stated that we'll see more lore regarding this at some point in the future, so we can look forward to seeing that.
So the long and short of the issue is; given what SE has said regarding interracial breeding in Eorzea, it's reasonable to make a half-breed character.
Also, while Eorzeans are typically a tolerant bunch, given how rare half-breeds are, it's up to you whether your half-breed character would face prejudice regarding their background.
Other considerations include what Roswyn has already mentioned; the significant physical differences between some of the races make certain combinations... Improbable, at least until SE says otherwise.