I would suspect your character could be/play anything s/he desires. Linage doesn't seem to matter as much as RP. (Unless you are going for a half-giant arcane archer. Doable, but...wow.
Actually not do-able as a half-giant. You must be an elf (any subrace) or half-elf to qualify.My personal interpretation is that even a drop of Elvish blood (at least, enough to justify taking half-elf as your race) would be sufficient, however, I do not speak for the rest of the approvals team here.
Half-elf, half-half giant?
And in such if by Race they have some abilities, those would take many generations to disapear to, even if the blood was drawn thin. If those abilities are given by race then they are prominant genes, and knowing the frail nature of the Humans, those genes would take over the weaker genes.
While there might be something vaguely elven about a quarter-elf, I don't think there would be anything resembling even half-elven special abilities left, any more than a sorcerer can breathe fire.