A pointless undertaking.
The description 'fool' is now often used as a contemptuous insult, but in the Middle Ages it didn't have such negative connotations. A fool then was a naive simpleton but regarded with respect and even admiration - somewhat the way that 'the fool on the hill' is portrayed in The Beatles' song. The numerous names and phrases that contain the word 'fool' generally refer to how easy it is to dupe (or to fool if you like) a fool. Examples of this are:
Fool's gold - a brassy mineral that resembles gold.
Fool's paradise - a state of euphoria based on false hope.
Fool's parsley - Lesser Hemlock, a poisonous weed that resembles parsley.
Fool's mate - a naive chess move that incurs checkmate in two moves.
Fools rush in... - a proverb indicating the unworldly lack of caution shown by fools.
It has long been part of the initiation of new recruits to send them on 'fool's errands'. A credulous beginner might be sent to the stores to fetch a skyhook or a tin of striped paint. The first references to 'fool's errand' come in texts from the 18th century. An early example is from the Yorkshire-born clergyman Edmund Hickeringill's Priest-craft, 1705:
Did not the Pope send all the Princes in Christendom upon a Fools Errand, to gain the Holy Land, that he might (as he did in their absense) rob them of their territories.
Given that playing tricks on the simple-minded must have been happening since Adam was a lad, it seems odd that 'fool's errand' didn't emerge into the language until the 18th century. The reason for this is that mediaeval England had a different name for the sport, which was a 'sleeveless errand'. From the Tudor era to around the 1700s, 'sleeveless' was very commonly used to mean 'futile' or 'trifling'. 'Sleeveless answers' were those that gave no useful information and a 'sleeveless errand' was a fool's errand, often used to get someone out of the way. The historian Raphael Holinshed used the expression in Chronicles, 1577:
So as all men might thinke that his prince made small account of him, to send him on such a slevelesse errand.
'Sleeveless' had also been used for centuries before with the same meaning as now, that is, 'without sleeves', so it's reasonable to assume that's where the 'futile' meaning of sleeveless derived. What's not clear, and despite my best efforts I've not been able to find out, is why 'sleeveless' was used with that meaning. Such usage of the word has long since died out and, although it's not difficult to make guesses at the link between 'sleeveless' and 'futile', to know the real truth of that derivation we may need get aboard a time machine.