Getting a job with Uncle Sam -- whose $950 million-a-month civilian payroll is this area's major source of revenue -- is getting tougher every day.

Hiring has been frozen at eight federal agencies with nearly 20 percent of the 356,000 civilian government jobs here. Other agencies have drastically cut back their recruiting in anticipation of staff cuts required by the Gramm-Rudman-Hollings balanced budget law. Most agencies also are trimming expenses, ranging from travel to overtime.

The balanced budget law requires most civilian agencies to cut spending 4.3 percent by Sept. 30, while defense agencies must trim theirs by 4.9 percent. Even bigger across-the-board cuts will be imposed on agencies during the fiscal year that follows, unless Congress and the White House approve budgets that meet deficit reduction targets set up in the law.

The budget submitted by President Reagan, which must be approved by Congress, calls for about 23,000 federal job cuts in the fiscal year starting in October, but does anticipate growth in half a dozen defense and foreign affairs agencies.

At a meeting Wednesday of the Interdepartmental Placement Committee, made up of local federal hiring officials, it was announced that hiring has been frozen at the Veterans Administration (which has 6,600 employes here), National Aeronautics and Space Administration (4,800 local workers), Export-Import Bank (360 employes) and General Accounting Office (3,500 workers).

Also faced with a hiring freeze are the Justice Department (18,000 employes), Commerce Department (18,600) and General Services Administration (9,100 workers here). A hiring freeze had already been announced at the Department of Housing and Urban Development, which has 3,300 workers in this area.

The 5,200-employe Library of Congress has warned that up to 500 jobs may have to be eliminated by Oct. 1.

Agencies that have imposed limited hiring freezes, meaning that only "essential" workers will be replaced when they leave, include the Environmental Protection Agency, Federal Railroad Administration, National Institutes of Health and National Science Foundation.

Partial hiring freezes have been ordered at many Navy and Air Force installations, and other federal agencies are expected to join the freeze once they figure out the impact of Gramm- Rudman-Hollings on their operations.

The government hires between 8,000 and 10,000 people each month; about 14 percent of the federal work force is located in the Washington area.