"Virginia's state government seemed unable to find anyone to run the place who has not either applied boot polish to his face while at college or been accused of sexual assault. " What do "run the place" and "either applied boot polish to his face while at college or been accused of sexual assault" mean in the sentence?
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"Run the place" in this context means "Lead the government of the state of Virginia."
All of the available leaders have been accused of unacceptable past behavior: "applying boot polish" means applying something black to the face to mimic African complexion. "accused of sexual assault" is a legal term; assault can mean many things, including touching, inappropriate comments, threats, and demands.