At Fort Leonard Wood, Mo., lovingly remembered as the armpit of the world by countless thousands of Army trainees, government employes once ran the laundry at a three-year cost to taxpayers of $4.5 million.

When the job was handed to a private contractor in response to Office of Management and Budget directives, 95 federal jobs went to the big personnel roster in the sky and a three-year saving of almost $600,000 was realized.

The change, called contracting-out, was not without pain, however. That pain is at the center of a burgeoning battle between those who believe the government should not be doing work the private sector clearly can do and those who are either trying to save government jobs or who believe the private sector does not respond well to government, particularly military, needs.

The General Accounting Office recently studied the Fort Leonard Wood laundry and 17 other examples where the private sector assumed jobs once performed by government. At Fort Leonard Wood, "the contractor's performance was deficient in several areas during the first six months" but became satisfactory, GAO said.

In five other contracts, "unsatisfactory contractor performance was experienced" and new contractors were hired or problems were still being worked out at the conclusion of the GAO study. These kinds of difficulties are giving ammunition to those who question the contracting-out emphasis of the Reagan administration.

Officials at OMB argue that the difficulties experienced so far are management problems, reflecting inexperience or unwillingness on the part of government supervisors to define carefully the job they want done and then to make sure nongovernmental contractors do it. When management gets serious, they say, contractors do well, money is saved and all benefit.

The existing contracting-out directive (predecessors go back to 1955) was written in 1979, in the Carter administration, and contains this very Reagan-like sentence: " . . . It has been and continues to be the general policy of government to rely on competitive private enterprise to supply the products and services it needs."

That is not a popular stance with groups such as the American Federation of Government Employes (AFGE). Spokesman Gary Dinunno said contracting-out "abuses are so flagrant in some areas it's just obscene. We've had stories of illegal aliens being involved in security on military bases."

AFGE's lobbying efforts were instrumental in a one-year moratorium imposed by Congress on some contracting-out efforts by the Veterans Administration, but the big target for both AFGE and OMB is the Defense Department.

DOD purchases millions of dollars' worth of commercial services that are nongovernmental in nature, so many DOD operations are prime candidates for conversion to private enterprise. That means many government employes could find themselves off the federal payroll, although any contractor assuming a job once done by government employes is obliged to offer them a job first, though not necessarily with the same wages and benefits.

Thus, we find the House Armed Services Committee "concerned," as it said in the report on the Pentagon's fiscal 1983 authorization bill, "that the Executive Branch has lost sight of priorities in its headlong rush to contract out large numbers of federal jobs."

The report ordered a one-year moratorium on studies comparing the costs of public or private operation. Those studies are usually required as a prelude to conversion; as a general rule, if a private contractor can do it for 10 percent less, then the government is supposed to get out of the business.

There has also been some action in the Senate to limit contracting-out, but the issue is far from resolved. AFGE's Dinunno said, "I think we have some support" for limiting the practice.

AFGE does not know, he said, how many government jobs have been lost through contracting-out, but the Pentagon says that contracting-out so far has resulted in big benefits.

Lawrence J. Korb, assistant secretary of defense for manpower, reserve affairs and logistics, testified on Capitol Hill recently that, "We have conducted about 400 bidding competitions between government operations and private-sector contractors in the past three years. About 60 percent were won by private firms saving about $70 million a year. The other 40 percent of the competitions were won by in-house activities which, spurred by competition, became more efficient and cost-effective.

"As a result, we are saving an additional $14 million annually. So whether a contractor or the in-house activity wins the competition, the real winner is the taxpayer."

OMB is working on a new circular on how to go about contracting-out. The central emphasis is to reduce the paperwork required before competitive bidding can be held between the existing government office and would-be contractors, according to OMB officials. A draft will be published for comment soon.

An early draft, which OMB officials said was never official, proposed that functions employing fewer than 25 people could simply be turned over to the private sector without cost comparisons. That set the unions on edge and was part of the impetus behind the moratorium movement.

"The great assumption" of those favoring keeping jobs in-house, said Donald E. Sowle, OMB's administrator for federal procurement policy, "is that government does things perfectly, which is obviously not true."