I'm not sure if it helps at this point, but I've tried several "flavors" of apparently legit tar files with zero entries. All fail.

``tarfile`` module:

```
$ ( set -x ; cd /tmp || exit 1 ; python -V ; rm -fv test.tar ; python -c 'import os, tarfile ; fd = os.open("test.tar", os.O_WRONLY | os.O_CREAT | os.O_EXCL) ; f = os.fdopen(fd, "w") ; f = tarfile.open("test.tar", "w", f) ; f.close() ; f = tarfile.open("test.tar") ; print("okay so far; calling f.next()...") ; f.next()' ; openssl dgst -sha256 test.tar ; rm -fv test.tar )
+/bin/zsh:496> cd /tmp
+/bin/zsh:496> python -V
Python 2.7.13
+/bin/zsh:496> rm -v -fv test.tar
+/bin/zsh:496> python -c 'import os, tarfile ; fd = os.open("test.tar", os.O_WRONLY | os.O_CREAT | os.O_EXCL) ; f = os.fdopen(fd, "w") ; f = tarfile.open("test.tar", "w", f) ; f.close() ; f = tarfile.open("test.tar") ; print("okay so far; calling f.next()...") ; f.next()'
okay so far; calling f.next()...
Traceback (most recent call last):
  File "<string>", line 1, in <module>
  File "/Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/tarfile.py", line 2350, in next
    self.fileobj.seek(self.offset - 1)
IOError: [Errno 22] Invalid argument
+/bin/zsh:496> openssl dgst -sha256 test.tar
SHA256(test.tar)= 84ff92691f909a05b224e1c56abb4864f01b4f8e3c854e4bb4c7baf1d3f6d652
+/bin/zsh:496> rm -v -fv test.tar
test.tar
```

BSD tar (OS X):

```
$ ( set -x ; cd /tmp || exit 1 ; tar --version ; rm -fv test.tar ; tar -cf test.tar -T /dev/null ; python -c 'import tarfile ; f = tarfile.open("test.tar") ; print("okay so far; calling f.next()...") ; f.next()' ; openssl dgst -sha256 test.tar ; rm -fv test.tar )
+/bin/zsh:499> cd /tmp
+/bin/zsh:499> tar --version
bsdtar 2.8.3 - libarchive 2.8.3
+/bin/zsh:499> rm -v -fv test.tar
+/bin/zsh:499> tar -cf test.tar -T /dev/null
+/bin/zsh:499> python -c 'import tarfile ; f = tarfile.open("test.tar") ; print("okay so far; calling f.next()...") ; f.next()'
okay so far; calling f.next()...
Traceback (most recent call last):
  File "<string>", line 1, in <module>
  File "/Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/tarfile.py", line 2350, in next
    self.fileobj.seek(self.offset - 1)
IOError: [Errno 22] Invalid argument
+/bin/zsh:499> openssl dgst -sha256 test.tar
SHA256(test.tar)= 5f70bf18a086007016e948b04aed3b82103a36bea41755b6cddfaf10ace3c6ef
+/bin/zsh:499> rm -v -fv test.tar
test.tar
```

GNU tar (OS X via MacPorts):

```
( set -x ; cd /tmp || exit 1 ; gnutar --version ; rm -fv test.tar ; gnutar -cf test.tar -T /dev/null ; python -c 'import tarfile ; f = tarfile.open("test.tar") ; print("okay so far; calling f.next()...") ; f.next()' ; openssl dgst -sha256 test.tar ; rm -fv test.tar )
+-zsh:23> cd /tmp
+-zsh:23> gnutar --version
tar (GNU tar) 1.29
Copyright (C) 2015 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>.
This is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.

Written by John Gilmore and Jay Fenlason.
+-zsh:23> rm -v -fv test.tar
+-zsh:23> gnutar -cf test.tar -T /dev/null
+-zsh:23> python -c 'import tarfile ; f = tarfile.open("test.tar") ; print("okay so far; calling f.next()...") ; f.next()'
okay so far; calling f.next()...
Traceback (most recent call last):
  File "<string>", line 1, in <module>
  File "/Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/tarfile.py", line 2350, in next
    self.fileobj.seek(self.offset - 1)
IOError: [Errno 22] Invalid argument
+-zsh:23> openssl dgst -sha256 test.tar
SHA256(test.tar)= 84ff92691f909a05b224e1c56abb4864f01b4f8e3c854e4bb4c7baf1d3f6d652
+-zsh:23> rm -v -fv test.tar
test.tar
```

The discussion from #24259 does not appear to contemplate this case, and seems to imply an assumption that there will be at least one entry (which is not always the case).