Is iron the most stable element in the periodic table? According to the binding energy per nucleon vs mass number graph, it is observed that iron-56 has the maximum value of binding energy per nucleon ($\pu{8.75 MeV}$). It means that iron-56  is the most efficiently bound nucleus meaning that it has the least average mass per nucleon. This is the approximate basic reason why iron and nickel are very common metals in planetary cores since they are produced profusely as end products in supernovae and in the final stages of silicon burning in the stars. So, in one word, iron is quite stable.  



But, what about helium and other noble gases? They are considered the most stable elements in the whole periodic table. But their binding energy per nucleon value is less than iron-56. So, they are not stable as iron-56. Is that true? Is iron the most stable element in the periodic table both structurally and chemically?
Yes, $^{56}\ce{Fe}$ has the most stable nucleus, and $\ce{He}$ is the most chemically inert element. These are different and unrelated qualities, pretty much like physical fitness and intelligence in a man. As for structural stability, there is no such thing in chemistry (there is one in architecture and another in mathematics, but those are out of scope of this question).