The etymology of 局
Upper part of 局 represents 尺 (ruler), alluding to rules and canons.
Lower (inside) part of 局 represents 口 (mouth), alluding to unbounded freedom.
口 (mouth) is under (inside) 尺 (ruler): 局 implies a sense of constraint.
局 is a pictogram of 博 (Liubo), an ancient Chinese board game. The relationship between 局 and 博 (Liubo) is mentioned vaguely: "簙有局以行十二棊". 局 can either mean a round of the game or the board on which the game is played.
The etymology of 局面
盘中排营寨城池，眼前无弓箭旌旗。心内有刀枪剑戟，局面儿几般形势。『元 周德清 《斗鹌鹑·双陆》曲』
局面 first appeared in a Qu written by 周德清 in Yuan dynasty. It refers to the situation of an ongoing Backgammon game.
Please Note: My answer only considers the origins of 局
. Their meanings change overtime and are impossible to encapsulate in an reasonably long answer. The "sitting with your back bent" hypothesis proposed by the questioner might indeed be true at some point in history.
- 20160423_0: Initial answer
- 20160423_1: Elaborated on the 2nd interpretation for 局 and left room for ambiguity