I will have a go at this since it came up once more.
To start with when we use relativistic equations we are in the realm of particle/nuclear physics, one way or another.
In the formula that puzzles you, $E = mc^2$, the mass is a variable too. So it is an equation that gives an equivalence of one variable to another. So as others have observed, this formula is a mathematical convenience, it follows the conservation laws that are imposed by the differential equations that describe all systems in the microcosm, and should absolutely hold.
Secondly the mass itself in your formula is the relativistic mass
and can be defined for all particles and particle systems whether moving or at rest:
$m_r=E/c^2$ or equivalently $m_r=\gamma\times m_0$
where $m_0$ is called the rest mass of a particle or a system and is an invariant quantity under four dimensional transformations, characterizing the Energy_momentum four vector of the system. $m_0^2=E^2-p^2$ where $E$ is the total energy of a system of particles and $p$ the total momentum.
When it is one particle under observation it is the mass that you will find in the table of masses for particles, and can be considered the building block of matter as we know it except for particles at the limit of $m_0=0$. The photon, for example ,has energy $E=h\nu$, but $m_0=0$
When a single particle is moving its relativistic mass rises according to the formula above, thus $m_r$ itself counted in units of $m_0$ is a continuous variable derived from the variables of the system under observation.
Now if we go to the bottom of what we know of the microcosm, we know that it consists by the particles given in the Standard Model table playing ball with each other according to very specific rules, called Feynman diagrams. They interact with each other throwing gauge bosons and turning into new particle manifestations with new $m_0$ masses conserving all necessary quantum numbers. There is never any energy floating by itself in the vacuum. It is always carried by a particle according to the formulas given above.
So the answer for the microcosm is, in a nutshell, that neither $m_r$ nor energy are composed of anything. Only $m_0$ exists and the rest are mathematical definitions useful for calculations. Since the macrocosm is made up by the particles in the microcosm this holds true for the macrocosm too, though relativistic concepts are irrelevant for the everyday velocities achievable .