I am sure Elrond is half-elven, so Arwen is too. I recall something saying Argorn was half-elven as well. Shouldn't half-elves not have to become mortal to marry each other?
Saying that someone is "half-elf" is a way of saying that they have an elven parent and a mortal one. Elrond fits into this category (his mother, Elwing, was an elf; his father Earendil was a human). But one is not thereby "half-human, half-elf" - one has to choose between being an elf and being a human:
Thus, at the time of the War of the Ring, Elrond is quite definitely an elf. Arwen gets that same choice.
But Aragorn is quite definitely a human; he's a descendant of Elros. Thus, he is going to die; and Arwen makes the choice to die as well, that is, to be under the Doom of Men, in order to be with him. It is only at the very end of her life that she sees what's involved with this and regrets it:
She doesn't seem to ever reconcile herself to it.
I doubt she had to; rather she decided to.
There's precedent for a full-blooded (immortal) Elf marrying a mortal, where the Elf retains (or seems to retain) their immortality:
There doesn't appear to be any requirement that an immortal must become mortal in order to marry a mortal; in fact, they can't. Lúthien is the only full-blooded Elf who did this, and she had to die and be reborn. Only half-Elves of the line of Eärendil were given the mortal choice.
Having said that, by all accounts Arwen decided to give up immortality; bear in mind the words she uses when professing her love to Aragorn:
And what Tolkien writes in Letter 153:
Notably, he frames her choice in terms of Lúthien, not in terms of mortality/immortality. I'll get to why I think this is important in a moment.
The question then becomes: why did she make the choice?
Tolkien's remark above gives one clue; "the choice of Lúthien" is described thus in The Silmarillion:
If we take the Professor at his word, then one of Arwen's motives is revealed: she's so in love that she would rather die than spend the rest of eternity without him.
As well, though there's no evidence she was thinking along these lines, I've commented before that the nature of the Eldar is such that there are practical benefits to not marrying beneath your mortality:
But that's perhaps reaching for explanations; I tend to agree with Matt Gutting that she wasn't thinking that far ahead.
1 There's actually in-universe speculation that Tuor was granted immortality, though personally I find it doubtful; I'm particularly intrigued by Tolkien's statement in Letter 153 that "'it is supposed' (not stated) that he as an unique exception receives the Elvish limited 'immortality'"
As half Elves, Elrond and his brother Elros had to choose between being mortal men or being immortal Elves. Elros became a man, Elrond became an Elf.
So it isn't that loving Aragorn itself requires Arwen to become a mortal; staying in Middle-earth after Elrond departs will make her become a mortal.
As for Aragorn being a half-Elf, he isn't. See the answer to this question to see just how little Elven blood ran through Aragorn's veins. He was something like 0.00000000000000000677% Elf.