Why do organisms accumulate potential energy? I can understand that animals need some battaries to run. But, we learn that plants serve like batteries for animals because they accumulate the sun energy in the first place! You can eat them or burn to extract the accumulated energy. This means that they consist of structures that they want to blow-up and disintegrate to fall apart at the lower energy level. But why? 
This is strange to me since the most fundamental law of nature says that battaries spontaneously leak and discharge themselves. This means that the hydrocarbon structures that all life organisms are made of on the Earth are not stable. The photosinthesis, $\mathrm{H_2 O + CO_2 + \text{photo} \mapsto O_2 + CH_2O}$, is costly. It not only needs (photo) energy but uses it to produces the hydrocarbons, the $\mathrm{CH_2O}$-like spontaneously degrading structures! Why do you want to consist of something unstable that has tendency to blow up as soon as possible (why not immediatly, BTW?) if your purpose is survival, as any life organism on the Earth?
I watched the Gibbs Free Energy but could not understand anything. Is it because of the min. energy law we have everything at the min energy already on the Earth and if you want to produce some structures (fortunately, there is a Sun), you have no other choice but to make them unstable?
 A: There are many things on Earth that do persist over time by being made of thermodynamically stable substances. A common example is a good solid lump of granite. These can "survive" unchanged for billions of years (much longer if it wasn't for plate tectonics). However, we do not say they are alive.
There are a much smaller number of other things that persist over time in a different way: by using energy to build up a thermodynamically unstable structure, and then later breaking down that same structure to get the energy back, which is then used to capture more energy and build more structure, etc. Living organisms are a good example of this, although it also applies to other things, such as fire, thunderstorms and rivers. The general term for these is "dissipative structures", because they persist by downgrading, or "dissipating" energy.
Of course it would be nice to have the best of both worlds, where you're made of a thermodynamically stable substrate and you're also able to grow by extracting energy from your environment. I guess a growing crystal would be an example of this in a sense - but the problem is that thermodynamically stable structures tend not to be very interesting. It would be impossible to find a thermodynamically stable structure that could extract energy from sunlight to build more of itself, because as soon as it's absorbed a photon and converted that energy into a chemical bond, it's out of thermodynamic equilibrium and therefore unstable. In general the more complex a structure is on the molecular level, the lower its entropy tends to be. If you want to be more complicated than, say, a rock, you don't really have much choice other than to be a far from equilibrium structure, which means you contain stored energy.
You're right that all batteries leak. If you've decided to persist by being a living organism, that's just something you have to put up with. As a human, most of the energy you extract from the food you eat will go into maintaining your cells against the continual forces of decay that degrade them, with only a much smaller proportion going into useful things like growth and movement.
A more prosaic answer is that life doesn't seem to have much of a choice about what it's made of. All life on Earth is made of water, proteins, and smaller amounts of lipids and other organic compounds. At the origins of life this probably wouldn't have been so far from equilibrium, since in a reducing atmosphere amino acids can form spontaneously. However, the combined result of photosynthesis and limestone formation is that the atmosphere is now highly oxidising, which means that on today's Earth protein burns rather nicely.
In summary, the reasons that living organisms are made of unstable matter are that (i) if they didn't extract energy from their environment, we wouldn't consider them alive; (ii) if you extract energy from your environment, you become thermodynamically unstable; (iii) it's difficult if not impossible to have a complicated, yet organised, molecular structure without being far from equilibrium; and (iv) the decision to be made of proteins, which are particularly unstable, was made a long time ago, when they weren't anywhere near as flammable as they are today.
