As with many of these kind of words in English, you have to look at the origin and family history:
From the Online Etymology Dictionary:
their (pron.) : plural possessive pronoun, c. 1200, from Old Norse þierra "of them," genitive of plural personal and demonstrative pronoun þeir "they" (see they). Replaced Old English hiera. As an adjective from late 14c.
there (adv., conj.) Old English þær "in or at that place, so far as, provided that, in that respect," from Proto-Germanic *thær (source also of Old Saxon thar, Old Frisian ther, Middle Low German dar, Middle Dutch daer, Dutch daar, Old High German dar, German da, Gothic þar, Old Norse þar), from PIE *tar- "there" (source also of Sanskrit tar-hi "then"), from root *to- (see the) + adverbial suffix -r.
Meanwhile "they are" also comes from the Old Scandinavian þeir, þer, or þair (the root of "their"), and Old English earun. Apparently the Norse "they" replaced the Old English hīe used at the time, but people still used the Old English "to be" verb. The contraction happened somewhere along the way between then and now.
As this comment states, they were probably pronounced very differently from how they are pronounced today, although "they" and "their" were probably always similarly pronounced since one is the root of the other.