Can the mass of an object account for the total internal energy of the object? While studying thermodynamics my book describes the quantity the internal energy of a system. My book states:

The concept of internal energy of a system is not difficult to understand. We know that every bulk system consists of a large number of molecules. Internal energy is simply the sum of the kinetic energies and potential energies of these molecules. We remarked earlier that in thermodynamics, the kinetic energy of the system, as a whole, is not relevant. Internal energy is thus, the sum of molecular kinetic and potential energies in the frame of reference relative to which the centre of mass of the system is at rest. Thus, it includes only the (disordered) energy associated with the random motion of molecules of the system. We denote the internal energy of a system by $U$.

The question that comes to the mind that can we measure the internal energy of the system? Yeah I know that in systems containing large number of particles this would be quite a tedious job to do. But I have learned that mass of a system is the measure of the energy contained in a system. So I thought the following:
For a body at rest which is far from any celestial body (i.e., free from their gravitational influence) 


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*Can we say that the mass of the body is due to the total internal energy of the system (i.e., thermal energy,... etc.) and that if we can measure the mass then we know the value of the total internal energy of the system? i.e., does $E_{int} = m_0 c^2$? If not why not? 


Given that we have a all the required instruments to measure the mass precisely (i.e., to high degree of precision).  Here $m_0$ is the mass of the object. 

The problem arises because my chemistry textbook in the thermodynamics chapter says that we cannot measure the total internal energy of the system. 
 A: Thermodynamics is a classical theory, and also a theory that has been demonstrated to be emergent from statistical mechanics. 
$E=mc^2$ is a formula of special relativity, which involves Lorenz transformations between systems moving with velocity close to the velocity of light. It is a misleading formula because the $m$ is a variable depending on velocity, and velocity is not an invariant quantity.
Thus your suggestion is not physically logical.
The mass of an object is called the invariant mass, and does not change between inertial frames. It is defined by the sum of the four vectors of all particles making up the system.
Summing the four vectors of a solid body will include the kinetic energies of the particles, but the kinetic energies cannot be simply extracted from the invariant mass of the system, because it is total energy that is invariant in  an inertial frame, not the kinetic energies.
A: This has been discussed in detail in an earlier question on this exchange.  See Can mass-energy equivalence be used to measure absolute internal energy?.
A: 
But I have learned that mass of a system is the measure of the energy
  contained in a system.

It is not clear whether or not your question applies to internal energy per classical thermodynamics, or if was intended to include mass-energy equivalence per $E=mc^2$. My answer below is from the classical thermodynamics perspective.
Since internal energy is an extensive property, the amount of internal energy that a given object possesses is indeed proportional to its mass. But that is not the same as saying two objects of identical mass have the same amount of internal energy. The simplest example is two identical objects with different temperatures. The masses of the objects are identical, but clearly the internal kinetic energy component of one of the objects is greater than the other. 
Hope this helps.
A: Not sure what the problem here is, but as your book clearly states, with you even emphasizing the correct line, internal energy is NOT rest-mass energy!
The rest-mass energy of an electron for example is $E_0=m_0 c^2 = 511 keV$, while a set of electron in thermodynamic equilibrium at 300K temperature have on average $0.025 eV$ kinetic energy.
Even in a relativistic regime, when the kinetic energy would dominate the rest-mass energy, $\gamma m_0 c^2\gg m_0 c^2$, where $\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is the relativistic factor, you still couldn't perform the measurment you propose, because you would need to know the velocity or velocity distribution, i.e. temperature for that.
A: If you include the mass of the particles in the internal energy then it is identical to the mass of the system up to a factor c$^2$.
This total energy can in principle be measured by determining the mass by for example weighing. In practice it will be very hard because kinetic and potential energy of the individual particles making up the system are so much smaller then their masses. In nuclear systems however the so-called mass defect is significant.
