What are the etymologies of 局 and 局面?
The etymology of 局


1st interpretation


  促也。从口在尺下，復局之。『說文解字』
  促也。以㬪韵爲訓。从口在尺下復局之。尺所以指厈規榘事也。口在尺下。三緘其口之意。『說文解字注』


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.

2nd interpretation


  一曰博，所以行棊。象形。『說文解字』
  一曰博所㠯行棊。象形。博當作簙。簙、局戲也。六箸十二棊。簙有局以行十二棊。局之字象其形。此別一義。『說文解字注』


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 and 局面. 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.

Source 


http://www.zdic.net/z/18/sw/5C40.htm
http://baike.baidu.com/view/240375.htm



Revision History


20160423_0: Initial answer
20160423_1: Elaborated on the 2nd interpretation for 局 and left room for ambiguity