Saving water... as in storing drinakable water correct? Not saving water by reducing your consumption.
Storing water.... we have 6 gallon jugs in our deep freezer, helps fill up the space, thus making it work less to keep things frozen, and will provide cooling if the freezer is without power for a few days.
We also have a few cases of bottled single use water, stacked up next to the dryer. We rarely drink it ourselves, but keep on hand for "emergencies" / grab and go from poor planning on our part when we are running out the door.
Other than that, we don't store any additional water. If city water stopped working there is a good sized creek/ small river a few blocks away. We could haul water from there and filter with our Berkey water filter.
The Berkey is our back up plan. We could also set up a water catchment system with a bit of work and wait for rain and then filter that.
The Berkey is great! Love it, we've even taken it camping and pulled water from our creek side river and filtered it. The Berkey was created for missionaries traveling 3rd world countries back in the late 1800's. It get out everything! We have both black filters (standard - parasites, bacterial, etc) as well as the white filters (additional order - take out floride, cloramine and all the "modern" water additions the gov't puts in).
I know some folks keep "oil drums" of water in the basement and rotate them. A drop of bleach will keep them "fresh". Also, don't forget that if you've got a water heater in your residence, there is another 40 gallons of water just waiting to be drained.
Plan on each family member needing a gallon a day, not all of that would be for drinking, but also washing hands, etc.