The nuclear waste problem that has built up over 35 years of nuclear weapon-making in the United States is 25 times larger than that of the entire American atomic power industry, and it's expected to get even bigger.

In giant steel tanks at its Savannah River facility, the Department of Energy holds 25 million gallons of high-level radioactive waste, the by-product of production reactors that have been turning out plutonium and other nuclear materials for 28 years.

The waste is increasing at a rate of about 1.5 million gallons a year, and Savannah River officials expect that rate to go up to almost 2 million gallons a year with a projected major buildup in nuclear weapons over the next eight years.

All the nation's nuclear power plants have created only 3 million gallons of similar high-level radioactive waste since the first one opened, according to DOE estimates.

And Savannah River's waste is only a part of the nation's defense waste. By the end of last year, there were 76.8 million gallons of high-level radioactive defense waste temporarily stored here and at two other DOE facilities.

The radioactivity level of the waste stored at Savannah River has been measured at 3,000 rads per hour, a rad being a unit of measurement for radiation. Half the people exposed to a level of 450 rads per hour to the whole body can be expected to die, according to currently recognized scientific data.

DOE is running out of space for building new tanks, and some of the older ones here have developed leaks. So the agency has begun to develop ways of compressing the volume of waste and is searching for a permanent, safe, underground storage site.

A billion-dollar nuclear waste processing plant is being designed for construction at Savannah River. It would combine the highest level waste, now sitting in giant tanks in the form of brown sludge, with a special form of glass. That glasslike mixture would then be poured and sealed in stainless steel cylinders that will be 2 feet in diameter and 10 feet long.

By combining the sludge, which is made up primarily of strontium 90 and slight amounts of plutonium 239, with glass, the volume of the waste would be sharply reduced and solidified in a permament form that scientists believe will never leak -- even over the thousands of years during which it will remain radioactive.

This glassification process is similar to one the French have had in operation for several years.

If the new facility is completed as expected in 1989, it will take 12 years of operation, turning out 500 cylinders a year, to catch up with Savannah River's nuclear waste backlog and output.

Even then, the waste problem would only be halfway solved.

For one thing, the solidified nuclear waste would be even more radioactive than it had been in its liquid form, according to officials here.

"At the surface of the cylinder," one scientist said during a recent interview, "the radiation will be close to 20,000 rads an hour."

Because of this high radiation level, the cylinders initially produced in the waste processing plant will have to be stored in specially constructed steel-and-cement buildings at Savannah River. That building will be cooled, because of the small amount of heat the radioactive material releases, and operated by remote control because of the danger of exposure to any humans.

But Savannah River will only be able to store two years' output of cylinders, according to officials.

The DOE plan calls for creation in the early 1990s of another billion-dollar facility, a permanent "depository" for the glassified waste, which is not expected to lose its high level of radiation for thousands of years.

A test facility has been constructed 1,500 feet below the surface at the Nevada Test Site to try out concepts for the permanent depository. There, spent fuel from power plants has been sealed away within a granite rock formation. Scientists are attempting to determine if the below-ground, man-made caverns can retain the heat and radiation from this nuclear power plant waste.

If the caverns work out, the cylinders with Savannah River's glassified waste eventually would be transported to such a repository for disposal.

Meanwhile, however, the liquid waste now stored and being generated at Savannah River has to be shuffled among a series of carbon steel tanks surrounded by thick cement retaining walls and covered up to their tops by dirt.

Nine of the original tanks have shown cracks, and one actually had leaked radioactive cesium through its steel wall and the cement protective shell. That tank has since been emptied.

The first cracks were noticed 10 years ago, and a multimillion-dollar program to build 27 new tanks was begun. Thirteen have been completed, each with cement walls several feet thick and two rather than just one carbon steel shell.

Hundreds of cooling rods lace the interior of each of these 1.3 million gallon tanks, for the waste as it comes from the plutonium processing plant is not only radioactive, but as hot as the surface of a 200-watt bulb. It takes two years, sitting in such a tank, before the high-level radioactive sludge sinks to the bottom and the water rises to the top.

But the musical chairs shifting of radiation within the tanks won't be able to handle the expected increase in waste, and even now it is known that at least eight additional new tanks will have to be built.

Savannah River's waste problem is only half that of DOE's other major nuclear reactor production facility at Hanford, Wash. There, tanks are also leaking but money has yet to be allocated for some permanent waste disposal approach.