The Fairfax County Board of Supervisors gave its support yesterday to a lawsuit to force the Virginia Department of Corrections to take 70 state prisoners out of the seriously overcrowded county jail. The jail opened in 1978 with a capacity of 254, but currently holds 361.

It was not clear whether the state could take the 70 prisoners -- all of them convicted felons with more than six months to serve -- even if Fairfax won the suit.

"We're trying to take every single prisoner we can," said state corrections spokesman Wayne J. Farrar, "but we just don't have any room." Farrar said the state has only 7,990 general-purpose beds, but he is currently housing 8,365 inmates by utlizing other space, such as infirmaries.

Furthermore, Fairfax Board Chairman John F. Herrity, while supporting the suit, said Virginia Gov. John N. Dalton had told him, "There is no realistic solution" to the prisoner problem.

The state has 500 additional beds in facilities under construction, and has long-term plans for 2,000 more, but the Correction Department estimates that by 1990 there will still be a shortage of more than 4,000 beds because of the steadily growing prisoner population. The post-World War II baby boom and improved law enforcement and cited as major reasons for the growing prisoner population.

While Fairfax wants the state to take 70 convicted felons off its hands, the state wants Fairfax, along with other localities, to create more diversion programs to stretch the capacity of existing prison and jail facilities.

Fairfax applied for state funds under the Community Incentives Diversion Act, passed by the General Assembly in 1980, but it didn't get a grant because the Board of Supervisors voted to exclude from the program those convicted of buglary, breaking and entering, and drug-related charges.