Old English stæf (plural stafas), "walking stick, strong pole used for carrying, rod used as a weapon, pastoral staff," probably originally *stæb, from Proto-Germanic *stab- (source also of Old Saxon staf, Old Norse stafr, Danish stav, Old Frisian stef, Middle Low German and Middle Dutch staf, Old High German stab, German Stab, Gothic *stafs "element;" Middle Dutch stapel "pillar, foundation").
This is reconstructed to be from PIE root *stebh- "post, stem, to support, place firmly on, fasten" (source also of Old Lithuanian stabas "idol," Lithuanian stiebas "staff, pillar;" Old Church Slavonic stoboru "pillar;" Sanskrit stabhnati "supports;" Greek stephein "to tie around, encircle, wreathe," staphyle "grapevine, bunch of grapes;" Old English stapol "post, pillar").
As "pole from which a flag is flown," 1610s. In musical notation from 1660s. Sense of "group of military officers that assists a commander" is attested from 1702, apparently from German, from the notion of the baton that is a badge of office or authority (a sense attested in English from 1530s); hence staff officer (1702), staff-sergeant (1811). The meaning "group of employees (as at an office or hospital)" is attested by 1837.
Staff of life "bread" is from the Biblical phrase break the staff of bread meaning "cut off the supply of food" (Leviticus xxvi.26), translating Hebrew matteh lekhem.
The Old English word, in plural, was the common one used for "letter of the alphabet, character," hence "writing, literature," and many compounds having to do with writing, such as stæfcræft "grammar," stæfcræftig "lettered," stæflic "literary," stæfleahtor "grammatical error," with leahtor "vice, sin, offense."