About 3/4 of a gallon.
A ten-gallon hat doesn’t hold 10 gallons. If it did, it would be pretty unwieldy. It actually would hold roughly 3/4 of a gallon or 3 quarts.
So, why the heck is it called a 10-gallon hat?
The name is believed to originates from a mistranslation.
“Cattle drivers and ranchers in Texas and the Southwest often crossed paths with Mexican vaqueros who sported braided hatbands—called “galóns” in Spanish—on their sombreros. A “10 galón” sombrero was a hat with a large enough crown that it could hold 10 hatbands”Why do we call it a 10-gallon hat? – HISTORY
American cowboys adopted the name to describe their own sombrero-inspired headgear as “10-gallon hats.”