The District government unintentionally held a bizarre type of lottery last week: The grand prize was cheap housing, the number of frustrated hopefuls averaged 35,000 an hour for a week, and the odds of being a winner wouldn't pass muster at any track.

The District, holding only 300 federally funded certificates for low-income housing, decided that the only fair -- and safe -- way to distribute them was to take participants on a first-call, first-served basis.

So, in a city where there are 13,000 families on a waiting list for public housing, officials advertised three telephone numbers to be called only from 9 a.m. to 4 p.m. for the five work days last week.

The response could modestly be called overwhelming.

In the first two hours, the telephone company recorded 110,000 phone calls to the District housing department's 535 exchange. With three ring-overs on three lines, only nine calls could be taken at one time.

All others -- that is, almost everyone calling -- got a busy signal or a recording that the circuits were busy and to call back.

And call back and call back they did. Night and day, to all numbers at the housing department, to the offices of City Council members, to the mayor, to the media. But most of them failed to hit the winning combination in the telephone roulette.

"They had all of our offices tied up," said Lenox Elmore, chief of the department's Section 8 division. "Every agency in the city has gotten some calls . . . . Our phones are still continuing to ring."

By the end of the week, 3,927 persons had been registered, according to the D.C. Department of Housing and Community Development. If eligible for the program, most of those will go on a waiting list.

Only the first 300 eligible persons who got through actually will get housing certificates in the federal Section 8 existing housing program for low-income persons.

The nine phone answerers fielding the calls recorded the exact time of each to make sure they were put on the list in proper order, Elmore said.

The last time the city took names for the Section 8 housing lists was in 1979, and the long lines outside city offices created "near-riotous conditions," according to housing department officials who yesterday justified the method they used this time.

"Handicapped and elderly people were being pushed around. It was an ugly scene," Elmore recalled of the 1979 registration.

"I don't know that there could have been any more fair a way of giving everybody a chance to get at what we have," he said. It did not appear that callers from any particular area of the city had a better chance of getting through than others, he added.

While acknowledging that only a handful of those trying managed to get connected, Elmore said the department is not considering reopening the rolls. "It's closed," he said of the waiting list.

Under the Section 8 program, which has been drastically cut back under the Reagan administration, families pay one-third of their income for an apartment and the program pays the rest of the rent.

There are income limits, such as $17,900 for a family of four in the District, and rent limits, such as $440 a month for a two-bedroom apartment here, Elmore said.

One District resident, Anita Wilson, said she and a number of friends tried continually and none got on the list. Home with a cast on her arm, Wilson said she now lives with her parents and was frustrated all week in her constant attempts to apply for the Section 8 program.

"I tried to call [Elmore], but his line was busy, too," Wilson said.