The Reagan administration's first attempt to reduce the size of the federal government -- a total freeze on the hiring of federal employes, retroactive to Nov. 5 -- has caused havoc in some government agencies, created agonizing hardships for hundreds, perhaps thousands, of individuals and put an enormous new bureaucratic burden on the Office of Management and Budget.

The new administration obviously hopes that the hiring freeze will have some positive effects as well, both as a symbol of its determination to cut the size of government and as a means of saving federal dollars.

According to informed sources, the White House opted for the retroactive freeze as much for its shock value as for its substantive impact. The new White House team hopes to replace the freeze relatively quickly with specific new personnel cutbacks to be imposed through the budget process, these sources said. They added that even if some exceptions to the freeze are forced on them, other people will be permanently discouraged from trying to take up the jobs they've been offered.

But the benefits won't be evident for some time, and the difficulties the freeze has caused began to show up immediately. The new administration's first executive order managed to unite such disparate parties as federal personnel officers, union leaders, conservative Republican and liberal Democratic members of Congress -- all of them united against the freeze. A round of court challenges to the freeze started yesterday, and many agencies have already begun the process of appealing for exceptions. Congressional offices have started to protest, and affected would-be federal employes are advertising their tales of woe.

This furor represents a substantial achievement for the first days of the new administration. And it is a hint of what may lie ahead if President Reagan sticks to his budget-cutting guns and moves from a relatively unpopular target like the federal bureacracy to more popular ones like college loans and food stamps.

Thomas J. Brahan had never been to Washington before he stepped off the Amtrak train at Union Station last week, fresh out of the University of Wisconsin with a degree in industrial engineering.

But he had heard "nothing but good things" about life in the nation's capital, so he had chosen a career with the federal government over offers from five major private employers in the Midwest.

In his pocket he carried a letter that confirmed his hiring by the U.S. Patent and Trademark Office as an examiner, G85, Step 10, at a salary of $15,947 a year. Brahan had been offered the job Dec. 4, after being interviewed on the campus in Madison Nov. 17.

Brahan, 24, found a place to live through a roommate referral agency, did a little sightseeing over the weekend and found his way early Monday morning to Room 9-C 05 in Crystal Plaza Building No. 2 in Arlington.

There, Brahan and nine other would-be patent examiners sat patiently in the personnel office until about 11:30 a.m., "when we were called in by a Miss Monica Young, who told us we didn't have jobs. She explained what had happened and promised she would work hard to try to get us on the hardship roll. If she succeeded, we would be among the first on it."

Before the dejected Brahan and the others left Crystal City, they met with Alan Douglas, president of the Patent Office Professional Association, who told the dumbfounded young engineers and lawyers that his independent union would try to help get them on the payroll.

In the meantime, Brahan is "staring at the four walls," sitting among his not-yet-unpacked possessions in his newly rented, sparsely furnished Alexandria townhouse. The landlord agreed to wait for the rent until Brahan and his two roommates got their first paychecks from their new federal jobs. But one of them already has returned to his home in Schenectady, N.Y., and Brahan, down to his last $100, isn't sure how he can come up with his share.

"I'm going to give it until Friday, and then start looking for something else," he said yesterday.

The new freeze on federal employment will create bureaucratic red tape and absorb many thousands of working hours in the Office of Management and Budget just at the time budget examiners there will be under maximum pressure to find new ways to cut back government spending programs to fulfill the pledges of the new administration.

According to informed sources at OMB, career officials there are privately distressed at the prospect of devoting their energy to dealing with appeals from the freeze. But that is what they will have to do as requests for exceptions are passed on to OMB for consideration.

And there will be numerous requests, even if Reagan's appointees to the Cabinet impose a hard line in their departments. The Justice Department, for example, seems certain to appeal for an exception for 114 young lawyers recently offered jobs under the department's "honors program" for new graduates of law school and clerks for judges around the country. These 114 -- winners in a seeepstakes that involved 2,550 applicants -- received formal offers to begin work sometime later this year.

The honors program is a key element in Justice recruiting. The officials responsible for it feel strongly that if the jobs offered under the program this year are withdrawn because of the Reagan freeze, recruiting topflight new law graduates in future years will become virtually impossible, because future recruits will be wary that they might not be able to rely on the department's job offer.

The Internal Revenue Service is likely to plead that reductions in its workforce will cost the government money by diminishing its ability to audit tax returns or pursue tax-evaders in court. (The IRS, too, has just offered jobs to a large number of lawyers, offers that the freeze would nullify.)

At the National Highway Traffic Safety Administration, career officials are preparing an appeal to preserve the job of the man they selected (after a nationwide competition involved 121 applicants) to head the Heavy Duty Vehicle Research office. The man who got the job works for a major auto maker in Detroit, but he has formally resigned his job, put his house up for sale and taken other steps to move to Washington. "This is the best man we could find and we won't be able to keep him if the freeze sticks," a senior official said.

A class-action challenge to the freeze was filed in U.S. District Court here yesterday by the National Treasury Employes Union. It contests the new president's right to impose a hiring freeze retroactive to the presidency of his predecessor, whose own hiring freeze was in place at the time.

Though exact figures are not available, it is likely that the federal government has offered jobs to about 20,000 people since Nov. 5, the date to which the Reagan freeze was made retroactive. Some of them are already on the job, and the freeze will not apply to persons who have begun work. But thousands more find themselves in positions like Thomas Brahan's.

Ten of them were supposed to start training today in El Centro, Calif., to learn how to be patrolmen for the U.S. Border Patrol. All 10 jobs were canceled, including the one George Britton was to take.

Britton had sold all his furniture, left his wife in Fort Smith, Ark., and taken a plan to California to begin his training. Instead, he ended up spending last night with Bill King, chief of the El Centro border patrol station. "The border patrol takes care of its own," King told a reporter yesterday, though Britton was not one of its own and may never be.

According to W. Bowman Cutter, who ran three hiring freezes for the Carter administration as a senior offical of OMB, most of the people affected by the freeze will not be Washington bureaucrats, but will be people like Britton who expected to work for the National Park Service, the Forest Service, the Border Patrol and the Customs Service. From his experience, Cutter said in an interview, the workers that will be most affected in Washington are secretaries.