don't mind AI search results, there are a number of job categories that must work as they are essential, but won't get paid until the government is back up and running. (As you pointed out, the bills don't stop rolling in.):

Essential jobs that work without pay during a shutdown

National security and law enforcement

Military personnel: All active-duty members of the Army, Navy, Air Force, Marine Corps, and Coast Guard continue working without pay during a shutdown.
Federal law enforcement: Agents from agencies such as the FBI, CIA, DEA, and Secret Service remain on the job.
Border security: Border Patrol agents, Customs and Border Protection (CBP) officers, and Immigration and Customs Enforcement (ICE) agents continue their duties.
Federal prisons: Correctional officers must continue to report to work.

Air travel

Air traffic controllers: They remain on the job to manage air traffic.
Transportation Security Administration (TSA) agents: TSA officers continue to staff airport security checkpoints.

Public health and safety

Medical staff: Employees at Veterans Affairs (VA) medical facilities and other federal healthcare personnel, particularly those providing inpatient and emergency care, are required to work.
Disaster response: Staff from the Federal Emergency Management Agency (FEMA) continue to respond to emergencies.

Infrastructure and critical operations

Social Security and Medicare: While recipients still receive their benefits, some staff work to ensure basic functions continue, though services may be limited.
IRS employees: In recent shutdowns, contingency plans have required some IRS employees to work, at least temporarily.
Power grid maintenance: Workers responsible for maintaining the power grid are considered essential and must continue their work.

Who is not affected by the shutdown

U.S. Postal Service employees: Since the USPS is an independent agency funded by its own revenue, its employees continue to be paid during a government shutdown.
Members of Congress: They continue to receive their paychecks during a shutdown due to their constitutionally protected compensation.

The uncertain fate of federal contractors
Unlike federal employees, contract staff who work for federal agencies are not guaranteed back pay after a shutdown ends. Many contractors, including janitorial staff and support personnel, are often sent home without pay and may never be compensated for lost wages.