Was there a time period in human history when obesity was not rampant anywhere in the world, and what kept it from happening? ~

Before electricity and automobiles there was lots of work and exercise. 

Yep, before the widespread easy accessibility of sugar, high processed carbohydrates and take-away foods.
Where ever they are, within 20 years more than half the population is dangerously overweight.

We've banned cigarettes. Let's ban added sugar and white flour for the same reasons.

I think obesity comes with too much of everything.  

Look in history where food and wine and lust were plentiful and you will find loads of obesity. 

Look in history where people are eeking out an existence with little. and hardships aplenty, and you won't find any.  I just bet you there were no fat slaves in the South unless they were maybe the ones with the unfortunate label of  HOUSE darkies. The others were not fat.  The fiield hands.  They were probably bone thin.    I bet the Jews in Nazi Germany were very thin themselves.   I bet there were no fat cavemen.

I bet there are still few fatties in Ethiopia.