These are from PIE *swesor, one of the most persistent and unchanging PIE root words, recognizable in almost every modern Indo-European language (Sanskrit svasar-, Avestan shanhar-, Latin soror, Old Church Slavonic, Russian sestra, Lithuanian sesuo, Old Irish siur, Welsh chwaer, Greek eor). French soeur "a sister" (11c., instead of *sereur) is directly from Latin soror, a rare case of a borrowing from the nominative case.
According to Klein's sources, probably from PIE roots *swe- "one's own" + *ser- "woman." For vowel evolution, see bury. Used of nuns in Old English; of a woman in general from 1906; of a black woman from 1926; and in the sense of "fellow feminist" from 1912. Meaning "female fellow-Christian" is from mid-15c. Sister act "variety act by two or more sisters" is from vaudeville (1908).
Old English byrgan "to raise a mound, hide, enclose in a grave or tomb, inter," akin to beorgan "to shelter," from Proto-Germanic *burzjan- "protection, shelter" (source also of Old Saxon bergan, Dutch bergen, Old Norse bjarga, Swedish berga, Old High German bergan "protect, shelter, conceal," German bergen, Gothic bairgan "to save, preserve"), from PIE root *bhergh- (1) "to hide, protect." Meaning "cover, conceal from sight" is from 1711. Related: Buried; burying. Burying-ground "cemetery" attested from 1711. Buried treasure is from 1801.
The Old English -y- was a short "oo" sound, like modern French -u-. Under normal circumstances it transformed into Modern English -i- (in bridge, kiss, listen, sister, etc.), but in bury and a few other words (merry, knell) it retained a Kentish change to "e" that took place in the late Old English period. In the West Midlands, meanwhile, the Old English -y- sound persisted, slightly modified over time, giving the standard modern pronunciation of blush, much, church.