What does "their very existence" means here?
With their very existence dependent on the successful production of food, farmers had little, if any, time for removing stumps and stones.
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One of the definitions of very (#6 at the link) is as a synonym for actual, which is how it is being used here.
This serves as an intensifier. It's not their existence as farmers, but their actual existence as living people that was dependent on food production. These farmers had to successfully produce food or die of starvation.