Q:

Cost-effective alternative to rechargeable batteries?

Batteries are known to be the most unsustainable and expensive part of system exploiting renewable energy.
So I am wondering, are there any alternatives to storing energy in batteries? Here I am especially interested in household power supply.
How Stuff Works lists a few:

One of the oldest techniques people have used is the falling weight. You lift the weight to store the energy in it and then let the weight fall to extract the energy. Many grandfather clocks and cuckoo clocks use this technique. By running the string attached to the weights through a gear train, you can use a heavy weight and let it fall over a long period of time. (See How Pendulum Clocks Work.) This approach doesn't work very well in an electric car, but it has worked well in clocks for hundreds of years.
Many power plants use the "falling weight" approach in the form of water. The water is pumped uphill to a lake at night when the power plant has excess capacity. During high-demand daytime periods, the water runs through a turbine on its way downhill to a lower lake.
Another way to store energy is in some form of repeatable mechanical deformation. This is the idea behind a spring used in a wind-up clock or a rubber band used in a wind-up airplane. You store the energy by bending (deforming) the material in a spring, and the material releases the energy as it returns to its original shape. At the scale of a car, this technology has problems because of the weight of the spring, but at smaller scales (like a wristwatch) it works great. See also this page for an interesting example.
Nature has been storing energy for a long time, and if you want to think about it in this way, gasoline is really a form of stored energy. Plants absorb sunlight and turn it into carbohydrates (see How Food Works for a discussion of carbohydrates). Over millions of years, these carbohydrates can turn into oil or coal. On a more human time scale, we burn wood (which is a carbohydrate) to release stored energy, or turn corn into alcohol and burn the alcohol.
Another technique that nature uses to store energy is fat, which many of us are familiar with in a personal way. It is interesting to think about a car that somehow eats grass or some other carbohydrate and stores it as fat!
You can take energy and split water into its hydrogen and oxygen atoms using electrolysis. By storing the hydrogen and oxygen in tanks, you can later create energy by burning it, or (more efficiently) by running it through a fuel cell (see How Fuel Cells Work).
You can use the energy to spin up a flywheel and then later extract the energy by using the flywheel to run a generator. This patent has lots of information.
You can store heat directly and later convert the heat to another form of energy like electricity. This page discusses some of NASA's experiments in this area.
You can use compressed air to store energy. Toys like the Air Hog store energy in this way. Compressing gases like nitrogen enough produces liquid nitrogen, and this page talks about how you can use liquid nitrogen to power a car.
One of the new technologies that may become available in the future involves antimatter. When you combine normal matter with antimatter, you get energy. You store the energy by creating the antimatter. This page talks about it a bit.

I find the compressed air or water an interesting idea. Is it possible to compress air or water during the day using photo-voltaic power and then generating power at night by decompressing air or water? Water especially can be interesting in combination with rainwater harvesting. But is that efficient? Are any of the above more efficient and cost effective than batteries?

A:

Constraints
There are several aspects to energy storage, that can make one form better than another for a given set of circumstances.
Cost is usually the biggest driver. Energy can be stored at pretty much any scale you like, and cost is typically the constrainer.
There are additional constraints for within-household energy storage: in particular, safety, space and noise. Safety and space will rule out most forms of energy storage: high-energy-density solutions tend to be prone to bad accidents, precisely because they have high energy density.  And low energy density solutions need a lot of space, which is typically limited in households.
Electricity is hard
Storing electricity, to use as electricity, is a pain at the household level. The constraints above, rule out most of the energy storage solutions proposed in the question. It either takes too much space, or is too highly concentrated, creating a major safety risk.  There are some forms of highly concentrated energy that we have proven means of safe storage: but the capital costs for the round trip from electricity to storage to electricity are high, and the efficiencies are low, making the whole exercise very expensive.
These constraints rule out pretty much all the suggested methods in the question.

Towards a solution: start with the biggest energy uses
First look at your biggest energy uses. They are likely to be either heating, or cooling, or both. The good news is that they are easy to store, as low-grade heat or coolth, for 24-48 hours. Possibly more, if you can build some large underground storage.  Rich ancient Romans managed year-round thermal storage with large underground ice houses and heat stores.
Your biggest energy uses are also the place to start looking for energy efficiencies too. If you're planning energy storage, then the business case for efficiencies starts looking much better, because you get two sets of savings: the reduced energy costs, and reduced cost of your energy storage capacity.
Virtual storage
If the timing of energy use can be shifted, then that acts as virtual storage, with 100% efficiency. It may also require little capital investment - maybe a timer or two.
Storage efficiency
Efficiency in and of itself isn't necessarily a meaningful number: it has a direct impact on costs, which do matter. But in its own right, higher is not always better. If all other things are equal (and they rarely are), then higher is generally better.
Forms of energy
Not all forms of energy are equal. Some, such as low-grade heat or coolth, are easy to store, but can't do much work. Others, such as electricity, are capable of doing a lot of work (electricity is a high-exergy vector), but are very expensive to store directly (supercapacitors). Converting between forms of energy is always less than 100% efficient, and can be expensive.  Similarly, DC electricity and AC electricity are different: conversion each way is less than 100% efficient, and requires dedicated equipment that can be bulky, expensive, noisy, etc. Batteries convert DC electricity to chemical energy for storage, and back again for use; but most household appliances run on AC.
One sketch of a solution

Do as much energy-efficiency measures as are practicable: LED lights yielding over 100 lumens per watt, lots of thermal insulation, high-efficiency white goods, making the most of natural ventilation, solar gain, and night-time cooling.

run all your heating through a large thermal store, which you charge up whenever energy is readily available. Do the same, as far as possible, with a large ice store, for your cooling needs.

Find someone who's converted their household electricity distribution system to DC as much as possible, and has worked out which devices can be run direct off DC. Then copy their successes, and learn from their failures. Find the cheapest chemical batteries (probably lead-acid car batteries as of 2013), and use for round-trip electricity storage. If you have to, get a rectifier and an inverted to do the DC/AC two wa conversion too. I don't know if this is at all practicable, but it looks possible on paper, so someone mut have tried it.

