So the Federal government is limited and can only legislate in certain specific areas. The states can legislate more broadly, having what is known as general police powers, as long as they don't violate a right granted in the Constitution or conflict with federal law, including treaties. But the SCOTUS has long recognized that no right is absolute and can be abridged if three criteria are met.
- A law is necessary to a "compelling state interest";
- that the law is "narrowly tailored" to achieving this compelling purpose;
- and that the law uses the "least restrictive means" to achieve the purpose.
None of these quarantine orders have been adjudicated in court yet, but there is relevant history allowing for it.
As for the federal responsibility, they can do things such as restrict travel between states and across the border, regulate the purchase and sale of relevant supplies across state lines etc. There is an argument that Congress can pass a law using it's ability to regulate interstate commerce to give the President the right to order businesses to close, but I'm doubtful that the same law could allow the President to forbid the state from closing them, because that would run into federalism issues. Another thread to talk about this would be fun imo.
