Two years ago, Virginia officials estimated that 55,000 families in the state lived in houses with no running water or flush toilets. The effort to remedy the problem is just getting underway.

"Anybody who doesn't have a bathroom in 1990 isn't doing it on purpose," said Chuck George of the Virginia Water Project. "This is the first time we can really get to the problem."

A two-year, $5 million grant fund was set up by the last session of the Virginia General Assembly to install plumbing in some houses, although it may only put a dent in the problem.

"Our estimate is that, after we have had two successful years, we will have improved 1,000 units," said Warren Smith, associate director of the state housing department. "If you look at that compared to the 55,000, it's really not that big."

Smith and housing activists plan to ask the state for more money in two years, basing their request on the improvements made between now and September 1991.

"We want to show them that the people who have been saying, 'We can fix the problem' can actually do it," said George, whose organization helps rural families build bathrooms and get connected to sewers.

It is difficult to pin down the problem, and the numbers are only approximate, but dwellings without bathrooms are found mostly in rural areas, particularly in pockets along the North Carolina border.

In Northern Virginia, Loudoun County is believed to have 471 houses without indoor facilities, Fauquier about 300, Fairfax 94 and Prince William 250.

The state will review applications for money from local governments and community organizations between now and September, when the program is expected to begin. Agencies may apply for up to $250,000, and $12,000 is the maximum that may be spent on a single house. Construction costs average between $8,000 and $9,000, officials say.