LIKE SO MANY of the people who come to it, St. Elizabeths Hospital is nobody's fond charge. It is an institution that neither the federal government nor the District of Columbia has decided how best to care for. As a result, when money is short all around, St. Elizabeths is left in a financial vise. And if this year's federal and D.C. budget actions take the most severe turns possible, the hospital could be forced to let hundreds of employees go and to cripple or eliminate important health programs. Neither federal nor city officials are predicting this worst-case situation, but the current financial dilemma is serious and does call for new attention.

It is this grim possibility that prompted a memo to St. Elizabeths from Robert L. Trachtenberg, deputy administrator for the federal Alcohol, Drug Abuse and Mental Health Administration, calling for a contingency plan to meet the worst possible budget shortages that might occur in the fiscal year that begins Oct. 1. The memo was not an "order" to fire anyone, Mr. Trachtenberg emphasizes, but merely a necessary response to the possibility that the hospital may come up $25 million short of what administrators had thought was necessary just to stay even with last year.

That is how the possibility of a 653-employee reduction came up, as well as other suggestions for curtailing or killing programs, Mr. Trachtenberg says. But however this year's budget comes out, negotiations over an eventual transfer of St. Elizabeths from the federal government to the District--which have been going on for years--should be stepped up.

For example, the vast majority of patients are from the District, and many could--and should--be cared for through other local programs, from nursing homes to alcoholic and drug treatment centers. To ease what would be a sudden severe financial blow to the city, the federal government could phase out its responsibility over a period of years and assist in financing this kind of care through grants or assistance to the city.

These ideas are neither novel nor inexpensive. The General Accounting Office, among other agencies, is considering what should be done. But until the governments involved agree on an arrangement for picking up the tab--as well as the responsibility for St. Elizabeths--it will continue to be an uncertain institution with unclear functions and not enough money to get better.