When I looked into it in my own case, it worked out to slightly under 1 kWh/day in standby losses, or about $2/month. I turned off the water heater when I was gone for a month once, and the savings do seem to have been in the noise somewhere. Assuming a Nest-like device to do that for me would cost the same ~$250 that a Nest does, it would take 20 years to recoup the cost of the device, even assuming it was so perfect that it saved all standby losses.
My current place doesn't have gas either...
And as others have pointed out I have a natural gas heater.
Yes. I've thought of it, but no, it is not a good idea.
1 - http://www.amazon.com/s/?ie=UTF8&keywords=rinnai+water+h...
Aside from the fact that you need the water to remain above a certain level or the water can become dangerous (e.g. legionella), what happens the night you play glow in the dark flag football and decide to take a shower? Or do a load of dishes? Etc.
The Nest "works" (in the absolutely marginal way that it does so) because it makes a middling difference. That would be a stark difference with a hot water tank.
My dishwasher contains a heating element and so does the clothes washer.
Put on a sweater? But yes, what you describe is exactly how many people with Nest thermostats end up disabling the "intelligent" functions.