How small could an orbital system be in our solar system? Answers to How does the Sun's gravity have so much force and pull on the solar system? How does it scale? Newton's law of gravity scales on orbits so that we can theorize very small orbits with very small masses.
Could two hydrogen molecules orbit one another or would external gravity prevent them from having a stable orbit? We know there are quite small orbital systems in our own asteroid belt, but is there a practical limit to how small an orbital system (with the little orbital system also orbiting the Sun) can be in terms of total mass of the orbital system?
If we only consider gravity, one answer may be found using the Hill sphere. This is the distance the gravity of a body dominates over the sun: $$r_H \approx a \left(\frac{m}{3M_\odot}\right)^{1/3}$$ where $a$ is the semi-major axis, $m$ the mass and $M_\odot$ the sun's mass.
Now, an actual body has some nonzero density $\rho$ and $m=(4\pi/3)\rho r^3$. If the Hill sphere is inside the body there will not be any orbits around it (they would be dominated by the sun's gravity). So, we get the equation $$r = a\left(\frac{(4\pi/3)\rho r^3}{3M_\odot}\right)^{1/3}$$ which simplifies to
$$\rho = \frac{9M_\odot}{4\pi a^3}.$$ Objects less dense than this have Hill spheres inside themselves: at 1 AU this density is  $4.3\cdot 10^{-4}$ kg/m$^3$ (a thin gas), while at 0.1 AU it is 0.4255 kg/m$^3$ - about a third of sea level air density.
For hydrogen atoms, if we calculate the density for a 25 picometer atomic radius I get a density of 25,570 kg/m$^3$ (in actual hydrogen gas the atoms are spread out way more). Hence there the Hill sphere argument actually allows them to orbit each other!
In practice this does not happen. The orbital period at (say) 3 atomic radii is $\sqrt{4\pi^2r^3/Gm}\approx 3.4$ hours and the binding energy is $1.5\odot 10^{-27}$ J. This is $4\cdot10^{-5}$ of the thermal energy of the cosmic background radiation: even if there wasn't any sunlight or other radiation from inside the solar system it would jostle the atoms enough that they would split.
This suggests an apparent way of answering the question: if the binding energy $Gm/r$ is less than typical disrupting energy the orbit will not be possible. Actually calculating the forces is non-trivial (there are many kinds, from Jupiter's gravity to solar heating) and weaker forces can sum up over time. Knowing the disruptive background also just gives an upper bound for $m/r$, one could have smaller orbits.
So the true answer will be given by how small dense objects we are willing to consider, and (as the other answer points out) local forces. In the solar system the most relevant may be electromagnetic charging due to the solar wind: if the objects are metallic and close they can even attract each other if they have the same charge (!). Things like magnetic fields, infrared radiation and solar wind will play a role, making the true answer somewhat undefined.