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By default, subprocess.Popen doesn't close file descriptors (close_fds=False by default) and so a child process inherit all file descriptors open by the parent process. See the PEP 446 for the long rationale.
I propose to partially backport the PEP 446: only make *private* file descriptors non-inheritable. The private file descriptor of os.urandom() is already marked as non-inheritable.
To be clear: my patch doesn't affect applications using fork(). Only child processes created with fork+exec (ex: using subprocess) are impacted. Since modified file descriptors are private (not accessible from the Python scope), I don't think that the change can cause any backward compatibility issue.
I'm not 100% sure that it's worth to take the risk of introducing bugs (backward incompatible change?), since it looks like very few users complained of leaked file descriptors. I'm not sure that the modified functions contain sensitive file descriptors.
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I chose to add Python/fileutils.c (and Include/fileutils.h) to add the new functions:
 /* Try to make the file descriptor non-inheritable. Ignore errors. */
 PyAPI_FUNC(void) _Py_try_set_non_inheritable(int fd);
 /* Wrapper to fopen() which tries to make the file non-inheritable on success.
 Ignore errors on trying to set the file descriptor non-inheritable. */
 PyAPI_FUNC(FILE*) _Py_fopen(const char *path, const char *mode);
I had to modify PCbuild/pythoncore.vcxproj and Makefile.pre.in to add fileutils.c/h. I chose to mimick Python 3 Python/fileutils.c. Tell me if you prefer to put the new functions in an existing file (I don't see where such function should be put).
File descriptors made non inheritable:
* linuxaudiodev.open()
* mmap.mmap(fd, 0): this function duplicates the file descriptor, the duplicated file descriptor is set to non-inheritable
* mmap.mmap(-1, ...): on platforms without MAP_ANONYMOUS, the function opens /dev/zero, its file descriptor is set to non-inheritable
* ossaudiodev.open()
* ossaudiodev.openmixer()
* select.epoll()
* select.kqueue()
* sunaudiodev.open()
Other functions using fopen() have been modified to use _Py_fopen(): the file is closed a few lines below and not returned to the Python scope.
The patch also reuses _Py_try_set_non_inheritable(fd) in Modules/posixmodule.c and Python/random.c.
Not modified:
* _hotshot: the file descriptor can get retrieved with _hotshot.ProfilerType.fileno().
* find_module() of Python/import.c: imp.find_module() uses the FILE* to create a Python file object which is returned
Note: I don't think that os.openpty() should be modified to limit the risk of breaking the backward compatibility.
This issue is a much more generic issue than the change #10897.

It looks like some of these do make the file descriptor accessible to the Python user. The following are from the RST documentation:
oss_audio_device.fileno()
oss_mixer_device.fileno()
epoll.fileno()
kqueue.fileno()
audio device.fileno()
I did not find any documentation for the "linuxaudiodev" module, but it also seems to supply a public fileno() method:
$ sudo modprobe snd-pcm-oss
$ python2
Python 2.7.11 (default, Dec 6 2015, 15:43:46) 
[GCC 5.2.0] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> import linuxaudiodev
>>> a = linuxaudiodev.open("w")
>>> a.fileno()
3
Unless there is demand for making these cases non-inheritable, it might be safest to leave them be. However it looks like the mmap file descriptor could be internal (though I’m not an expert on that module to be sure). So you might be okay with the mmap change.

> It looks like some of these do make the file descriptor accessible to the Python user. (...)
Oh right, I reverted changes on these modules.
What do you think of the patch version 2?
Changes since patch version 1:
* Revert changes in linuxaudio, oss, select, etc. modules
* Fix _Py_try_set_non_inheritable(): *set* the FD_CLOEXEC flag, don't clear it!
* Optimize _Py_try_set_non_inheritable(): avoid fcntl() if possible (backport issue #26770)

The patch looks reasonable as far as I understand it. I don’t know what the consequences of adding new PyAPI_FUNC() symbols to 2.7 is, or changing the Windows build files, so can’t really comment on those aspects though.
I left some questions about porting _Py_open() and _Py_dup().

Sorry, I loose interest on this backport. The patch is large, the risk of regression is high, the goal is non-obvious. I'm not sure that such bugfix is still useful since Python 2.7 is old and developers know how to workaround Python 2.7 limitations.
The problem of non-inheritable file descriptors is that it requires to modify a lot of code, not only a few functions.
For all these reasons, I prefer to close the issue as WONTFIX.