From the article:

  Sure enough, they found signs of wet ringwoodite in 
  the transition zone 700 kilometres down, which 
  divides the upper and lower regions of the mantle. 

From wikipedia:

  The transition zone is part of the Earth’s mantle, and 
  is located between the lower mantle and the upper 
  mantle, between a depth of 410 and 660 km.

So, if our planet is roughly 12,000 km in diameter (6,000 km to the center), then this region of depth represents a 100-200 km thick shell where pockets of water may exist. But let's max it out, and try to find the volume ((4/3) * (pi r^3)) using very rough, round numbers to get an upper ceiling.

Hypothetically, if it were a continuous shell of pure, fresh water encapsulating the entire inner mantle and core, and if it happened to be 200km thick, with an inner surface at 600 km deep (the shell's floor starts at a radius of 5,400 km) and an outer surface at 400 km deep (the shell's ceiling stops at a radius of 5,600 km), the intervening space could collect possibly a volume close to 76 billion liters of water.

Compare this to a hypothetical scenario, where our planet's rocky crust has a radius of 5,995 km, and is completely covered in one giant ocean of water 5 km deep (think Water World with Kevin Costner) for a total diameter of 12,000 km, this huge ubiquitous ocean would only contain 2 billion liters of water.

So, in a world of perfect spherical maximums, rounded to the nearest billions in liters, this subterranian zone could contain 38 times the water we currently observe on the surface of the earth.

You dropped some zeros. A billion liters is a cube 100 meters on a side, more like a drop than an ocean.

Wait, you're right. I think I dropped about maybe 4 zeroes off the exponential notation. Strike the "billion" and make it like 10^11th or 10^12th power. Either way, I think the ratios still work.

...and oh yeah, by the way, my units were the wrong order of magnitude; should've been kiloliters (not just liters), on top of the exponential notation.

It's not a continuous mass of liquid water either, though. It's mostly rock with water in between the grains.

It is not between the grains. It is in the crystal structure of the grains as OH-.

Drill baby, drill!