I have a fair idea of generating and modeling crystalline materials but what non commercial software programs can be used to model amorphous carbon of a specific dimension, say a $6\times6\times6$ Å cube filled with amorphous carbon?
I am no expert in this area, but just to get the ball rolling...
The main issue simulating disordered phases is that one typically needs large simulation cells to capture the relevant length scales associated with the disorder of interest. Large simulation boxes typically mean some compromise in terms of the computational method you can use. For example, using traditional DFT methods is limited to about 1,000 atoms.
In this context, in recent years there has been a lot of work on developing machine learning potentials (a very generic term) parametrized with DFT calculations, that give accuracies similar to those of DFT but at much smaller computational expense, enabling simulations like the ones you are interested in. To highlight an example, I will pick the Gaussian Approximation Potential (GAP) scheme, for which you can find the code here. I chose GAP for my answer because I recently read a paper using it that was directly looking at what you are interested in, amorphous carbon. You can find the paper here.
$6\times6\times6$ $Å^3$ is a very small size for an amorphous material, which supposedly has no long range order. But the minimal size that yields a usefull amorphous sample depends on the material and the property you want to study.
The difficulty usually lies in the generation of the amorphous sample(s) that you want to investigate. Here, the traditional approach is melt-and-quench (MQ), where you randomize the positions of the atoms with molecular dynamics (MD) well above the melting temperature of the material. Then the melt is cooled rapidly, cooling rates often are in the range of 1 to 50 K/ps. Here is a tutorial that walks you through the steps, for a commercial software though, but this is the only tutorial that I know of. You should be able to replicate the steps with any MD (force-field, semi-empirical, DFT) package that you like.
The MD can be performed with DFT for small samples and for short MD times, several dozen of ps at the most. Otherwise, the MQ can be performed with classical MD given the availability of a suitable parametrization or with a semi-empirical method such as DFTB. Afterwards, the sample can be further annealed or optimized, and investigated at any level of theory you desire.
As long as the software can handle a few dozen to a 100s of atoms you can use it to study amorphous materials.
What the other folks have said is valid. But just to answer your question specifically, if you want to do DFT, there are many free packages that can do this type of thing. If you are wanting to do AIMD, CP2k is very fast and can handle a lot of atoms (but it is really hard to compile!). I mainly work on lattice dynamics calculations and use ABINIT for this. ABINIT also can do a lot of different things with AIMD too, but I have never used it for this. I think CP2k will be faster for AIMD due to the algorithms they use, but I can't give any details without doing some googling. Some other free software that I have used and that work well are Quantum Espresso and GPAW.
If you only want to do classical molecular dynamics (i.e. many more atoms in a cell) you can try LAMMPS (very good) and Gromacs (haven't used it). I can't say whether or not the empirical potentials will be suited to your problem (this is an important physics question you will have to consider) but there are many options to choose from for carbon. Check the potentials that come with LAMMPS.
Lastly, it is possible to do hybrid quantum/classical simulations. I think these are hard to do correctly, but I have been told by some colleagues that they are a very powerful tool. You can calculate certain properties in the 'quantum' region while coupling this part to a classical simulation to try to recover some of the more long-range affects.
Hope this helps!