Why don't we use weights to store energy? One of the main reasons why we haven't switched to clean energy is the lack of efficient storage methods - But, why aren't we using dead weights to store energy and draw it back later when needed? As an example of what I mean:

 A: Let's spin some numbers to further illustrate the poor energy density of gravity-based storage systems. Assume that you have a 100 kilogram lead weight that you can lower into a 10 meter deep hole in your yard.
Now, how much energy can it store? This is given by potential energy formula $E = mgh$, thus $E = 100\,\mathrm{kg} \cdot 9.8\,\mathrm{m}/\mathrm{s}^2 \cdot 10\,\mathrm{m} = 9.8\,\mathrm{kJ} \approx 2.7\,\mathrm{Wh}$.
For comparison, a single AA-sized battery stores about $2\,\mathrm{Wh}$ of energy.
A: We do use mass in a gravitational field to store energy and have done for hundreds of years! 
Grandfather Clocks

Grandfather Clocks have used powering weights since the 1660s. This was when they first gained their tall thin shape. The weights seen in the picture slowly descend as their stored energy is released. In order to add energy back into the system the weight needs to be wound back up.
Ffestiniog Pumped Power Scheme

The Ffestiniog Power Station in Wales was opened in 1963 and was the UK's first large scale pumped hydroelectric energy storage system. The reservoir works in a very similar fasion to other grid storage solutions. When excess energy is produced (because coal power stations can't be shut down immmediatly for example) water is pumped from the lower reservoir back up to the higher one. When there is a surge in demand water flows in the reverse direction back down through turbines smoothing out any peak demand spikes making the whole grid more efficient.
GravityLight

Finally something very similar to the design in your question. Here is a product which recently went through an Indiegogo funding campaign. It's called GravityLight and works by the action of a falling weight to provide power to remote villages and others who live off-grid.
A: Another way to store energy in mass is the use of flywheels. You simply take a massive wheel and spin it up to store energy, use an electrical generator as a break to take energy out.
Their main limitation for use outside of the realm of professional settings where they can be properly monitored and maintained by experts is the potential to fail catastrophically. 
A: Other answers have noted why we don't use systems like this today, but here is another example of a gravity-based power generator that was used in the past.
I was unable to find a source or picture of this, so maybe someone else can help me out.  I'll update if I can find details. I know this info from visiting the Winchester House in San Jose, California and taking the tour.
In the late 1800's and early 1900's gravity-based power generators were used in the homes of some wealthy individuals, including the home of Sarah Winchester.  Somewhere beneath the home there would be a generator with a big crank, which would be connected to a platform with heavy rocks on it.  The chain would slowly lower and provide power to small electrical systems in the house.  A servant would go crank the generator to rewind the weight a couple times per day.
A: If anyone has ever owned or used a self-winding watch, then this technology which has been in existence at least since the 1960s, uses small weights to wind the watch forever, barring any mechanical breakdown. Seiko Watches made many such models in both men's and women's styles. I have one that was recently stolen that was in perfect working order, right up until the day that it was stolen. It is probably worth quite a bit of money, both as collector's item as well as a very well designed near-antique.
A: Another existing example of using weights to store energy is the Raised Weight Hydraulic Accumulator. Hydraulic power is used to raise a weight, which, when it falls pumps that hydraulic power back out again. A system like this used to be used to power London's Tower Bridge.
A: Gravel Energy Storage System describes systems using a bucket conveyor belt to move gravel up and down a hill. This is claimed to be more efficient and cheaper than pumped hydro, especially when a natural reservoir is not available.
A: We do. Just the weights we use are made not from lead, but from water. Many water reservoirs are also used to store energy by pumping water up when you have energy surplus, and letting it come down through the generators when you need energy. All you need for this purpose is two or more reservoirs at different altitudes.
A: The amount of heat produced for a given amount of kinetic energy lost to friction is tiny. For that reason, a weight can't give you anywhere near enough energy to heat up water very much. The reason objects heat up so little from friction is because their total thermal energy at room temperature is already really large because atoms around room temperature tend to move at about 300 m/s. We're actually lucky molecules have a tendency to move so fast because it also means chemical potential energy can be converted into quite a lot of mechanical energy. For that reason, I climbed the CN tower at a normal speed for a much shorter staircase the whole way without getting tired or stopping to rest at all and took 26m8s to do it. The chemical potential energy of reacting oxygen and sugar is actually a lot more per mass than 1/2((300m/s)^2).
A: We do, and we do it in the form of solid mass, too!

ARES uses rails to move concrete weights up and down.

 

Potential energy storage or gravity energy storage was under active development in 2013 in association with the California Independent System Operator. It examined the movement of earth-filled hopper rail cars driven by electric locomotives) from lower to higher elevations.

There is even an idea to use winches, as you described:

Stratosolar proposes to use winches supported by buoyant platforms at an altitude of 20 kilometers, to raise and lower solid masses.

Thus, your question is void, there is no reason we don't, because we do.
A: You can use dead weights, but you need a huge amount of weight.
For example the biggest pumped hydroelectric system in the world (the Gianelli Hydroelectric Plant in California, USA) uses water stored in a reservoir about 9 miles long by 5 miles wide, lifted through a height of about 300 feet. Even then, it can only supply about 5% of California's electricity usage for less than 2 weeks before running dry - and given the current long term droughts in California, it can't even do that, because there would be no water available to refill it.
Trying to build devices like this for individual homes would be hopelessly uneconomical.
One way to get "free" energy to pump the water is to use tidal barrages, but even in the most suitable locations, the amount of power you get from a given area of water behind the barrage is only the same order of magnitude as covering that entire area with solar panels. The biggest operating tidal barrage in Europe (which has been running for about 20 years) only supplies about 0.1% of France's total electricity consumption.
A: Indeed pumped storage, ARES, and Gravity Light are all examples of using weights to store energy but, like an earlier post suggested, the mass / vertical height required (and resulting cost) makes many examples prohibitively expensive. With the right geography, pumped storage is a relatively inexpensive way of storing large amounts of energy, but it only really makes sense at large scales because of many factors, including the pipeline infrastructure and the distance from typically remote mountainous areas to centres of demand (affecting the build cost and cost of transmission).
To provide large amounts of electricity storage capacity for a long period (e.g. 20 years or more of daily operation), technologies like pumped storage and compressed air are currently much cheaper than batteries, though they are constrained by geography.
Other electricity storage technologies involving weights include those being developed by Gravitricity, Gravity Power (shown below), and Ground-Breaking Energy Storage (effectively cutting a large cylinder of earth and then raising it by pumping water underneath).

We can also use buoyancy as a means of storing energy. Research at the University of Windsor in Canada is ongoing into buoyancy energy storage, where a buoy is pulled down towards the seabed using a winch to charge the system, then allowed to rise back up to the surface of the sea in order to discharge the system and generate electricity. The winch is connected to a motor-generator in just the same way as the pump-turbine in a pumped storage plant is connected to a motor-generator, and can be located at the water surface (either on nearby land or on some kind of platform/barge) and connected to the base of the buoy via a pulley. Others are working on similar technology, including a company called Buoyant Energy.

