As a general rule (i.e. in vanilla kernels), fork/clone failures with ENOMEM occur specifically because of either an honest to God out-of-memory condition (dup_mm, dup_task_struct, alloc_pid, mpol_dup, mm_init etc. croak), or because security_vm_enough_memory_mm failed you while enforcing the overcommit policy.
Start by checking the vmsize of the process that failed to fork, at the time of the fork attempt, and then compare to the amount of free memory (physical and swap) as it relates to the overcommit policy (plug the numbers in.)
In your particular case, note that Virtuozzo has additional checks in overcommit enforcement. Moreover, I'm not sure how much control you truly have, from within your container, over swap and overcommit configuration (in order to influence the outcome of the enforcement.)
Now, in order to actually move forward I'd say you're left with two options:
NOTE that the coding effort may be all for naught if it turns out that it's not you, but some other guy collocated in a different instance on the same server as you running amock.
Memory-wise, we already know that subprocess.Popen uses fork/clone under the hood, meaning that every time you call it you're requesting once more as much memory as Python is already eating up, i.e. in the hundreds of additional MB, all in order to then exec a puny 10kB executable such as free or ps. In the case of an unfavourable overcommit policy, you'll soon see ENOMEM.
Alternatives to fork that do not have this parent page tables etc. copy problem are vfork and posix_spawn. But if you do not feel like rewriting chunks of subprocess.Popen in terms of vfork/posix_spawn, consider using suprocess.Popen only once, at the beginning of your script (when Python's memory footprint is minimal), to spawn a shell script that then runs free/ps/sleep and whatever else in a loop parallel to your script; poll the script's output or read it synchronously, possibly from a separate thread if you have other stuff to take care of asynchronously -- do your data crunching in Python but leave the forking to the subordinate process.
HOWEVER, in your particular case you can skip invoking ps and free altogether; that information is readily available to you in Python directly from procfs, whether you choose to access it yourself or via existing libraries and/or packages. If ps and free were the only utilities you were running, then you can do away with subprocess.Popen completely.
Finally, whatever you do as far as subprocess.Popen is concerned, if your script leaks memory you will still hit the wall eventually. Keep an eye on it, and check for memory leaks.