It’s both. The Postal Service is a unique institution, with elements of both a business and a public service. As the Post Office Department, the Postal Service was originally created exclusively as a public service. In the early days, post offices and the creation of post roads made communication across the country possible. Taxpayer dollars funded its operations, like other federal agencies.
But, beginning in the 1970s, the Postal Service largely stopped receiving any taxpayer money. Postal operations are instead funded by sales to you, the postal customer. Now it depends almost entirely on sales to keep the lights on.
At the same time, it is still required to deliver to neighborhoods across the nation, 6 days a week, even those far-reaching places that private carriers don’t deliver to because it’s not profitable. If exclusively run as a business, the Postal Service probably couldn’t afford to deliver there either. But, as a public service, it is required by law to do so. And those private carriers that couldn’t otherwise afford to deliver to more remote destinations often rely on the Postal Service to get packages to you.