Popeye, Olive and even baby Swee'Pea are on a skiing trip in the French Alps. They're joined by a dog named Bernie, who despite his appearance, is evidently a St. Bernard. He even carries the requisite barrel of brandy around his neck (though it turns out to be something other than brandy).
Popeye, Swee'Pea and the dog ski down the mountain, but Olive refuses to join, afraid she might ruin her beautiful new raccoon coat. Olive goes skipping along the snowy terrain, picking a stray flower here and there when suddenly she walks behind a tree and emerges without her coat. Enraged, she follows a set of footprints that lead from the tree to a cave, saying "Grr! Grr!" every step of the way.
She rings the doorbell of the cave and then sneezes. A miniature avalanche covers her in snow. A mysterious hand from within the cave has a saw, which it uses to cut out a block of ice with Olive inside it.
Later, Popeye, Swee'Pea and Bernie look for Olive. They follow her to the same cave. Popeye rings the doorbell. Swee'Pea sneezes and ends up covered in snow like Olive. The mysterious hand hands Popeye a saw, and Popeye uses it to cut out a block of ice with Swee'Pea inside it. Popeye promises Swee'Pea he'll get him out, but the hand knocks Popeye over the head with a hammer and pulls him and the frozen Swee'Pea inside the cave. The unknown occupant slams the door in the face of Bernie, who pounds on it, trying to get in.
"The Big Sneeze" has some mildly clever humor. It turns out that the cave belongs to Quasimodo, the Half-Back of Notre Dame (his football helmet and jersey confirm it). He is the official echo in the area, which explains why everyone's echo comes back with a French accent. But most of the cartoon seems designed for very small children. Olive's growling about her coat is on the level of making faces for babies.