How can we calculate the total water mass that falls during a flood?
I don't have the answer for you, but I can point you to the tools that can be used to get such an answer.  The Advanced Hydrologic Prediction Service compiles rainfall grids using radar, surface observations, and a variety of other sources.  Options offered include daily, month-to-date, and archived data.

You can probably quite reasonably put the Shapefile download option into a GIS application, such as Grass GIS or ArcGIS, then limit it to the state (or area of interest) and compute an average or total.

Alternative, you can download it as NetCDF, which I know you'd be able to open in GrADS/OpenGrADS, mask it to a certain area using some sort of shape file or geometry, and then run an areal average on it.

I really do wish more straightforward, user-adaptable data were easily available (and no guarantee it's not, I just don't know a source).  But at least this will give you the tools to figure out.

And while you're focused on rainfall, perhaps read up on central California's Great Flood of 1852!



One other way to get an estimate of the total volume of water would be to use stream flow values combined with reservoir level changes, and attempt to estimate what percentage of rainfall escapes the reservoir system.  But at that point you're getting a very rough estimate, because the monitoring system doesn't measure every small stream emptying into the sea... and a significant (and varying) percentage of the water infiltrates the soil or evaporates.  Those complications could be incorporated into the calculation, but in the end, that seems a complicated way to solve a problem we already have more complete information on.

You asked for how much water that is.  Well, we can just as easily get a fair estimate from the map:


(month to date as of February 22nd)

If that's an average of 8" over the state of California, that's:
$ \pu{8 in} \times \pu{163,696 mi}^2 =$ 23 trillion gallons

Even if we're off by a factor of as much as 2 on the estimates, such that it's only 4" average, or alternatively 16" average, that just changes it to 11.5 trillion gallons and 46 trillion gallons.

Which begs the point... we don't often think in terms of trillions of gallons.  Trillions of gallons seems a ton, but even just a meager 0.35" over the state would equate to a trillion gallons.  Perhaps normalizing it to the area... such that it'd be $ \pu{140000 gal/mi^2}$... is a little better.  But still, a typical person would struggle to recognize whether any given value in this form would result in a flood or a droughts.

That's why we often just work in terms of percentiles, or graphs, or return periods, or just give the average in terms of inches itself.  It may not click great with everyone, but for many, such units are just a little more interpretable.

(And in case you wondered, a gallon of water weighs about 8.36 pounds, so that's about 100 trillion pounds of water, again sounding like a lot, but to most people, so does 1 trillion pounds of water from a little 0.10" inch rain system)