'Obsess' is Latin for 'to sit against' — a thought besieging your mind, kin to 'siege' via PIE *sed-.
To preoccupy or fill the mind of someone continually and to a troubling extent; to be constantly worried about something.
From Latin 'obsessus,' the past participle of 'obsidere' (to sit down before, to besiege, to blockade, to occupy), composed of 'ob-' (before, against, opposite) + 'sedere' (to sit), from PIE *sed- (to sit). The PIE root *sed- is one of the most productive in the entire family: from it came Latin 'sedere' (to sit), 'sessio' (a sitting), 'sedes' (a seat, abode), 'sediment' (that which settles — sitting particles), 'reside' (Latin residere, to sit back), 'preside' (Latin praesidere, to sit before), and 'president' (literally one who sits before); through Greek 'hezesthai' (to sit) and 'hedra' (seat, face of a geometric solid — hence 'cathedral,' from 'kathedra,' the bishop's seat); and through Germanic 'sittan' (Old English, to sit), producing 'sit,' 'set,' 'settle,' 'saddle' (a sitting device), and 'nest' (from Proto-Germanic *ni-sd-az, literally a sitting-down place, a place to settle into). To obsess was
The word 'siege' is a doublet of 'obsess' — both descend from Latin 'obsidēre' (to sit against). 'Siege' came through Old French 'siege' (a seat, then a military blockade), while 'obsess' came directly from the Latin past participle. A military siege and a psychological obsession are etymologically the same