Mary Jacob is 103 years old. She lives alone in her apartment in a high-rise building for senior citizens on 14th Street NW. Jacob proudly maintains her apartment almost singlehandedly, from her immaculately made bed to her shiny clean kitchen.

But Jacob "likes to stay to herself," said the building's administrator, Betty Ceasar. "She's kind of withdrawn when it comes to mingling with her peers." So Ceasar decided Jacob needed a friend, a woman with whom she could socialize. "Socializing lessens deterioration," said Ceasar. "Psychologically it's healthy."

For four hours Monday through Friday, Nellie Burt, 70, a member of the Senior Companion Program, visits Jacob. "Me and her talk about different things," says Jacob. "She pays attention to me because I pay attention to her . . . . We get along just like sisters."

Burt is one of 67 members of the companion program, a 3-year-old project that employs low-income senior citizens to provide other senior citizens with companionship or assistance with some aspect of day-to-day living.

The program currently serves about 150 elderly people, giving priority to those who are homebound. The service is free and available to all senior citizens regardless of income.

The program receives about 10 calls a day requesting the service, said the project's director, Dr. Ola Criss. "People are living longer, so there's more of a need each year," she said.

The duties of the companions vary according to the needs of the client. While Burt provides companionship to Jacob, other companions also help with housework, reading and letter writing.

Elihu Miles, 63, is the companion to Rufus Dent, a 65-year-old Northeast Washington man who recently lost his sight. Miles takes Dent for walks and helps him with shopping and trips to the barbershop.

"About the only thing the companions don't do is windows," said Norma Pinkney, the program's care-plan coordinator.

Each companion is scheduled to work 20 hours a week, helping one or two persons, but the companions frequently work longer hours with no additional pay. For example, Miles has made it a tradition to join Dent for holiday dinners and other special occasions. The two celebrated Father's Day together this year.

The program tries to develop strong relationships between client and companion rather than serve a great many clients on a less personal basis, Criss said.

To qualify for the program, the companions must have annual incomes between $6,000 and $20,000, depending on family size. They are paid $2 an hour and receive money for transportation expenses and one meal a day. Each companion must attend a 40-hour training program that includes classes in first aid, psychology and social behavior of the aged.

Criss said she would like to hire more companions, but the $230,000 in federal and local grants that finance the program will only pay for the current 67 companions plus a small administrative staff, she said.

Figures from the city's Office on Aging indicate that more companions are needed. In 1978, 12 percent of the city's 105,000 elderly citizens needed help with four or more of the 12 major tasks of daily living, which include dressing, housekeeping, grocery shopping and reading.

To raise funds to pay for more companions, the SCP has recently published a Senior Companion Cookbook that is now on sale for $3. The SCP plans to sell the cookbook through the mail as well as in university and local bookstores.

The companions contributed all the recipes, some of which they said originated more than 100 years ago and have been passed along by generations of families. Most recipes are from the South, reflecting the backgrounds of many of the senior companions who grew up in North and South Carolina.

The recipes range from the traditional, like macaroni and cheese and chicken and dumplings, to more unusual dishes, like oxtail stew, pickled watermelon rind and cracklin' corn bread.

Rufus Dent highly recommends the companion program. "People say they'll help you, but you can't count on them," he said. "People don't want to be bothered."

But his companion Miles, he says, is someone he can depend on. Smiling and taking Dent's arm, Miles says, "He's my main man, my main man."