"Running a still" refers to the illegal distillation of alcohol.
This dates back to Prohibition (1920-1933), when alcohol was banned in America in the hope of preserving and solidifying Christianity. But the consumption of alcohol was so widespread, the ban had the effect of outlawing everyday behaviour. People took to brewing their own alcohol at home. This home-distilled alcohol was referred to as 'bathtub gin' in northern cities and 'moonshine' in the rural areas of North Carolina, South Carolina, Georgia and Tennessee. Home-brewers would sell the alcohol to eager customers in towns and cities.