The largest desalination plant is the sun - with its action on the oceans and turning sea water into fresh water (by evaporation->clouds->rain) it has more capacity than any man made structure to pump ocean water. To "store" rain water would require the damming of a significant body of water - but you would have to tolerate a significant rise in the water level over that smaller region (Or if you can freeze the water, a very large pile of ice like Greenland).
For example if you built a 150 m tall dam at Gibraltar and kept all the water from the major rivers that supply it locked up, the rise in the sea you would need is over 140 m (2.5 million square kilometers vs 3600 million square kilometers for all oceans). Unfortunately the rate of evaporation of the Mediterranean is greater then the net flux of all the rivers supplying it so that would not work unless you covered the entire thing with a reflective plastic sheet to prevent evaporation. You would kill all the flora and fauna in the sea, and all tourism, too.
We need another plan to save the earth.