What is the most stable structure in our physical world?

Well, that depends a lot on what you include under the term "structure". Supermassive black holes may last about 10 to the power 100 years, before they evaporate - that's vastly more than the current age of the Universe. A galaxy as a whole will also last a long time, before it disintegrates (over a long, long time, many of the stars will escape; many others will fall into the supermassive black hole). Some particles such as protons can also last a long, long time (note that protons are made up of smaller particles, so you might consider that a "structure" as well).