The opening lines of The Old Man and the Seaplace the story somewhere in the Gulf Stream. From this, we can suppose that Santiago sets off to fish from somewhere in this area; from Cuba and the tip of Florida and on up the North Atlantic coast. Hemingway uses Spanish words throughout the text to indicate that Santiago and Manolin (Spanish names) live in a Spanish-speaking culture.
The old man was thin and gaunt with deep wrinkles in the back of his neck. The brown blotches of the benevolent skin cancer the sun brings from its reflection on the tropic sea were on his cheeks.
Tropic sea: the northern Tropic of Cancer runs between Cuba and Florida. All of these clues suggest that the story is set in Cuba. It isn't until about 12 pages in that there is a direct reference to this.
The successful fishermen of that day were already in and had butchered their marlin out and carried them laid full length across two planks, with two men staggering at the end of each plank, to the fish house where they waited for the ice truck to carry them to the market in Havana.
In this case, it helps to know some background information on Hemingway. He spent time in Key West and in Cuba. These experiences undoubtedly contributed to writing The Old Man and the Sea. So, the bulk of the story takes place off the coast of Cuba.