What it means
A fool's errand is a pointless mission you're sent on when there's no real chance it'll pay off. It's the sort of job that's dead on arrival, whether someone's winding you up or you're chasing something that was never gonna happen in the first place. Finding a shop that's been gone for years, getting a refund from a company that's ghosting you, same energy.
Usage examples
"Chasing that rumour was a fool's errand from the beginning."
"Trying to get a straight answer out of that call centre was a complete fool's errand, I hung up after an hour."
"Driving across town for a shop that shut last year turned out to be a right fool's errand."
"They sent the new apprentice on a fool's errand to fetch a left-handed hammer."
"Mate, trying to get that landlord to answer his phone is a fool's errand, save your breath."
Where it comes from
The phrase has been in English since at least the 18th century. It meant a useless or hopeless task, and it often popped up in prank culture too, where apprentices or juniors got sent off to fetch something impossible or nonexistent. That joke helped the phrase stick, but the bigger meaning is simply a waste-of-time mission.