The strong force, the Yukawa meson, gluons...
Enjoy this story, because it is the heart of the work that won Japanese physicist Hideki Yukawa the 1949 Nobel prize for physics.
A typical atomic nucleus has many protons. They are all positively charged and, at typical nuclear distances of 10-15 metres or less, they repel each other really strongly. The neutrons are neutral, so they don't help with the electric force. It is obvious that, within the nucleus, there must be some really strong attractive force, one that involves neutrons and protons. Further, it probably has a finite range because, once a nucleus gets more than 200 or so nucleons, it is not very stable: above this separation, electrostatic repulsion wins out. (See also the discussion of binding energy.)
A force with finite range? Doesn't that suggest a force mediated by virtual particles? So, they only travel a few times 10-15 metres which, at the speed of light, takes 10-23 seconds. Substitute that into the equation above and you get a mass of about 10-28 kg. This is much more massive than an electron (a light particle or lepton) and rather less massive than a proton (a massive particle or hadron). Let's call such particles mesons, for medium mass particles.
Medium-mass particles were found and Yukawa received his Nobel prize.
Particle physics has moved on, and virtual particles are now also used to explain the forces within nuclear particles. Protons and neutrons are made of (electrically charged) quarks. The attractive force between quarks is called the colour force and is transmitted by virtual gluons (for a while, particle physicists competed in inventing whimsical names) and, in analogy with quantum electrodynamics for understanding the atom, the new theory is quantum chromodynamics.
But to return to the title: without E = mc2, there would be no virtual particles, so no strong forces, so no attraction to hold nuclei together. The periodic table would have only one entry, chemistry wouldn't exist, animals made of biochemicals wouldn't exist and you wouldn't be here reading obscure footnotes.