Could an atom have a binary nuclei system? I can't seem to find this answer anywhere online. I am trying to work through a solid state physics textbook for research and this popped into my head. I don't have any mathematical background on quantum mechanics and other more advanced physics topics. 
Obviously, atoms can't be described using similar terms to a star system for the most part. Knowing that, I was wondering if having an atom with two or more orbiting nuclei would be a possibility. From my understanding, nuclei can really only be stable up to certain sizes which is why you get atomic decaying. Would there be a set of subatomic forces that might allow for this sort of equilibrium to occur?
 A: In star systems you have a lot of options for exotic systems like this (one famous example).  That flexibility comes about because star systems are large compared with the de Broglie wavelengths of any of their components, so that stellar orbits are not quantized.
With atoms and their electrons and nuclei you are further constrained.
As a commenter observes, if you have two nuclei and enough electrons to make things neutral, and remove thermal energy until particles are forced to bind to each other, what you get are molecules.  For example, in the dihydrogen ($\mathrm H_2$) molecule, the two protons are separated by roughly an ångstrom, which is very different from the $10^{-5}\,\mathrm Å$ that separates nucleons within a nucleus.  The $\mathrm H_2$ molecule has two electrons, which orbit both nuclei. The low-energy excitations of the hydrogen molecule are rotations, which have energy $E_J = \frac{\rm 15\,meV}{2} J(J+1)$ for orbital quantum number $J=0,1,2,\ldots$; these states correspond to different amounts of angular momentum as the two nuclei orbit each other.
A: Were you wondering if it was possible to have two nuclei close enough to each other so that the residual strong force holds sway, but just far away enough so that they don't merge together?  I know there are nuclei with different shapes, including prolate and oblate.
Some nuclei of the same isotopes can have different shapes, called isomers.

Isomeric decay -- some radionuclides exist in comparatively long-lived excited states known as "isomers." As an example, the isomer of 242Am has a half life of 141 years, but the ground state decays with a half life of only 16 hours. When an isomer decays, the excitation energy is usually emitted as a gamma ray.

Looking at the picture in this article, Plutonium-240 has an isomer that seems to be so prolate that it is (maybe) close to what you were asking about.
Nuclear Structure
Tantalum-180m is a great example of a nuclear isomer.  Ground state Ta-180 has a half life of about 8 hours.  But when this nucleus is in one particular metastable excited state, the half life is so long that this isomer is considered stable.
---Update---
Looking around a little, the term you are looking for is 'dinucleus`.
Dynamics of the dinucleus
From Dinuclear System Model for Dynamics and Structure, Conference Paper, July 2001, G.G. Adamyan et al.:

A dinculear system (DNS) is a configuration consisting of two touching nuclei which keep their individuality and exchange nucleons and/or clusters.  Other notations for a dinuclear system are quasimolecular configurations, nuclear molecules and bi-cluster configurations.

Molecules inside a Nucleus
Scholarpedia: Clusters in nuclei
A: Every nuclei is composed by some number of protons which are positively charged particles and thus repels each other apart. For gluing them together some number of additional neutral particles- neutrons -are needed for "amplifying" gluing attraction nuclear force.
Same problem holds for binary nuclei system,- these pair of nuclei will be repeling from each other due to electrostatic pushing force between composed protons. To hold them at same distance, exact opposite nuclear gluing force between nuclei is needed. However electric Coloumb potential and Yukawa (strong force) potential has different forms. By equating them together,- you could find a stable distance where these opposite forces compensates each other :
$$ \frac {e^2}{4\pi r} =\frac {g^2}{4 \pi r} \exp(-\mu r) $$
Now solving for $r$, gives :
$$ r_{\text{stable}} = - \frac 1 \mu \ln\left(\frac {e^2}{g^2}\right)$$
Next you can see from the Wikipedia a Yukawa potential comparison over Coulomb potential in the long ranges :

In the far-end Coulomb potential asymptotically approaches some fixed value, which is below Yukawa zero potential. In the low-end Yukawa potential quickly approaches infinity, while Coulomb potential - not so quickly and asymptotically too. They doesn't overlap.
Finally if such system could have theoretically some stable point in distance between pair of nuclei,- this still would produced unbearable problems due to Heisenberg uncertainty principle which states that no microscopic system can have precise position vector :
$$ \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~~ $$,
So a bit of distance increasing,- and these two nuclei will fly apart as a pair of high-speed shells. A bit closer - and this "nuclei dipole" will be crushed into "single thing" by a strong nuclear force.
So  overall conclusions is that NO,- a pair of separated nuclei can't be a stable COM system in an atom. Unstable, with huge decay rates ? Maybe.
