Old English sæd "sated, full, having had one's fill (of food, drink, fighting, etc.), weary of," from Proto-Germanic *sathaz (source also of Old Norse saðr, Middle Dutch sat, Dutch zad, Old High German sat, German satt, Gothic saþs "satiated, sated, full"), from PIE *seto-, from root *sa- "to satisfy." Related: Sadder; saddest.
In Middle English and into early Modern English the prevailing senses were "firmly established, set; hard, rigid, firm; sober, serious; orderly and regular," but these are obsolete except in dialect. The sense development seems to have been via the notion of "heavy, ponderous" (i.e. "full" mentally or physically), thus "weary, tired of." By c. 1300 the main modern sense of "unhappy, sorrowful, melancholy, mournful" is evident. An alternative course would be through the common Middle English sense of "steadfast, firmly established, fixed" (as in sad-ware "tough pewter vessels") and "serious" to "grave." In the main modern sense, it replaced Old English unrot, negative of rot "cheerful, glad."
By mid-14c. as "expressing or marked by sorrow or melancholy." The meaning "very bad, wicked" is from 1690s, sometimes in jocular use. Slang sense of "inferior, pathetic" is from 1899; sad sack is 1920s, popularized by World War II armed forces (specifically by cartoon character invented by Sgt. George Baker, 1942, and published in U.S. Armed Forces magazine "Yank"), probably a euphemistic shortening of the common military slang phrase sad sack of shit.