Old English cest "box, coffer, casket," usually large and with a hinged lid, from Proto-Germanic *kista (source also of Old Norse and Old High German kista, Old Frisian, Middle Dutch, German kiste, Dutch kist), an early borrowing from Latin cista "chest, box," from Greek kiste "a box, basket," from PIE *kista "woven container" (Beekes compares Middle Irish cess "basket, causeway of wickerwork, bee-hive," Old Welsh cest).
Meaning extended to "thorax, trunk of the body from the neck to the diaphragm" c. 1400, replacing breast (n.), on the metaphor of the ribs as a box for the heart. Meaning "place where public money is kept (common chest, mid-15c.) was extended to "public funds" (1580s). Chest of drawers is from 1670s.