HOLMES MILL, Ky.—The U.S. Postal Service plays two roles in America: an agency that keeps rural areas linked to the rest of the nation, and one that loses a lot of money.

Now, with the red ink showing no sign of stopping, the postal service is hoping to ramp up a cost-cutting program that is already eliciting yelps of pain around the country. Beginning in March, the agency will start the process of closing as many as 2,000 post offices, on top of the 491 it said it would close starting at the end of last year. In addition,...