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To reproduce the abort (the set_nomemory() function is being added to _testcapi in issue 30695):
$ ./python memerr.py
python: Objects/call.c:89: _PyObject_FastCallDict: Assertion `!PyErr_Occurred()' failed.
Aborted (core dumped)
The corresponding backtrace:
#0 0x00007ffff7131670 in raise () from /usr/lib/libc.so.6
#1 0x00007ffff7132d00 in abort () from /usr/lib/libc.so.6
#2 0x00007ffff712a45a in __assert_fail_base () from /usr/lib/libc.so.6
#3 0x00007ffff712a4d2 in __assert_fail () from /usr/lib/libc.so.6
#4 0x00000000004628ce in _PyObject_FastCallDict (
 callable=<built-in method excepthook of module object at remote 0x7ffff70ab8d8>,
 args=args@entry=0x7fffffffe330, nargs=nargs@entry=3, kwargs=kwargs@entry=0x0)
 at Objects/call.c:89
#5 0x0000000000426004 in PyErr_PrintEx (set_sys_last_vars=set_sys_last_vars@entry=1)
 at Python/pythonrun.c:641
#6 0x00000000004263ac in PyErr_Print () at Python/pythonrun.c:510
#7 0x0000000000427586 in PyRun_SimpleFileExFlags (fp=fp@entry=0x987620, filename=<optimized out>,
 filename@entry=0x7ffff6febcd0 "memerr.py", closeit=closeit@entry=1,
 flags=flags@entry=0x7fffffffe4cc) at Python/pythonrun.c:403
#8 0x0000000000427673 in PyRun_AnyFileExFlags (fp=fp@entry=0x987620,
 filename=0x7ffff6febcd0 "memerr.py", closeit=closeit@entry=1, flags=flags@entry=0x7fffffffe4cc)
 at Python/pythonrun.c:82
#9 0x0000000000437217 in run_file (fp=fp@entry=0x987620, filename=0x91a190 L"memerr.py",
 p_cf=p_cf@entry=0x7fffffffe4cc) at Modules/main.c:341
#10 0x0000000000437bae in Py_Main (argc=argc@entry=2, argv=argv@entry=0x918020)
 at Modules/main.c:901
#11 0x000000000041d7fa in main (argc=2, argv=0x7fffffffe6c8) at ./Programs/python.c:102

With pyerr_printex.patch we get the following correct results (rc=120 is set by Py_Main() after Py_FinalizeEx() returns with an error):
$ ./python memerr.py 5 12
set_nomemory(0, 5)
result = _PythonRunResult(rc=1, out=b'', err=b'MemoryError\n\nDuring handling of the above exception, another exception occurred:\n\nMemoryError')
=============================
set_nomemory(0, 6)
result = _PythonRunResult(rc=1, out=b'', err=b'sys.excepthook is missing\nMemoryError\n\nDuring handling of the above exception, another exception occurred:\n\nMemoryError')
=============================
set_nomemory(0, 7)
result = _PythonRunResult(rc=1, out=b'', err=b'sys.excepthook is missing\nMemoryError\n\nDuring handling of the above exception, another exception occurred:\n\nMemoryError')
=============================
set_nomemory(0, 8)
result = _PythonRunResult(rc=1, out=b'', err=b'sys.excepthook is missing\nobject : MemoryError()\ntype : MemoryError\nrefcount: 1\naddress : 0x7f7ae1be3eb0\nlost sys.stderr')
=============================
set_nomemory(0, 9)
result = _PythonRunResult(rc=1, out=b'', err=b'sys.excepthook is missing\nobject : \ntype : MemoryError\nrefcount: 1\naddress : 0x7f8aead15eb0\nlost sys.stderr')
=============================
set_nomemory(0, 10)
result = _PythonRunResult(rc=1, out=b'', err=b'sys.excepthook is missing\nobject : \ntype : MemoryError\nrefcount: 1\naddress : 0x7f7b489d1eb0\nlost sys.stderr')
=============================
set_nomemory(0, 11)
result = _PythonRunResult(rc=1, out=b'', err=b'sys.excepthook is missing\nobject : \ntype : MemoryError\nrefcount: 1\naddress : 0x7ff97365eeb0\nlost sys.stderr')
=============================
set_nomemory(0, 12)
result = _PythonRunResult(rc=120, out=b'', err=b"sys.excepthook is missing\nobject : \ntype : MemoryError\nrefcount: 1\naddress : 0x7f559c9e0eb0\nlost sys.stderr\nException ignored in: <_io.TextIOWrapper name='<stdout>' mode='w' encoding='UTF-8'>\nMemoryError")
=============================

pyerr_printex.patch LGTM, please convert it to a GitHub PR.

Is it good to ignore errors of setting sys attributes? Can we prevent these errors?

There are just 6 cases left where the return code of _PySys_SetObjectId() is ignored:
Python/pylifecycle.c|689 col 5| _PySys_SetObjectId(&PyId_stderr, pstderr);
Python/pylifecycle.c|1211 col 9| _PySys_SetObjectId(&PyId_stderr, pstderr);
Python/pylifecycle.c|1657 col 5| _PySys_SetObjectId(&PyId_stdin, std);
Python/pylifecycle.c|1666 col 5| _PySys_SetObjectId(&PyId_stdout, std);
./Python/pythonrun.c|104 col 9| _PySys_SetObjectId(&PyId_ps1, v = PyUnicode_FromString(">>> "));
./Python/pythonrun.c|109 col 9| _PySys_SetObjectId(&PyId_ps2, v = PyUnicode_FromString("... "));
Error should be handled there in the same way errors are handled within the functions where they are invoked (returning -1, NULL or aborting). This should be the purpose of another issue as it is acceptable I think that there is no test case for those changes.
I am leaving the day after tomorrow for few weeks, without internet access (sailing to the mid-atlantic), and will create then a new issue if no one else does that first.

I tried to set last_type, last_value and last_traceback to None at startup, and this also fixes a crash. But this changes behavior (these attributes became set), and deleting them doesn't guaranties that following setting will be successful, especially with new compact dict implementation.

Removing 2.7 as the problem cannot be reproduced here. PyEval_CallObjectWithKeywords() does not assert on PyErr_Occurred().

New changeset 66caacf2f0d6213b049a3097556e28e30440b900 by xdegaye in branch 'master':
bpo-30817: Fix PyErr_PrintEx() when no memory (#2526)
https://github.com/python/cpython/commit/66caacf2f0d6213b049a3097556e28e30440b900

New changeset d5d79545b73110b2f4c2b66d150409514e2ca8e0 by xdegaye in branch '3.6':
[3.6] bpo-30817: Fix PyErr_PrintEx() when no memory (GH-2526). (#4107)
https://github.com/python/cpython/commit/d5d79545b73110b2f4c2b66d150409514e2ca8e0