diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index 395c0f1d300..18d415adf54 100755
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -1604,17 +1604,20 @@ class TarFile(object):
             # Find out which *open() is appropriate for opening the file.
             def not_compressed(comptype):
                 return cls.OPEN_METH[comptype] == 'taropen'
+            error_msgs = []
             for comptype in sorted(cls.OPEN_METH, key=not_compressed):
                 func = getattr(cls, cls.OPEN_METH[comptype])
                 if fileobj is not None:
                     saved_pos = fileobj.tell()
                 try:
                     return func(name, "r", fileobj, **kwargs)
-                except (ReadError, CompressionError):
+                except (ReadError, CompressionError) as e:
+                    error_msgs.append(f'- method {comptype}: {e!r}')
                     if fileobj is not None:
                         fileobj.seek(saved_pos)
                     continue
-            raise ReadError("file could not be opened successfully")
+            error_msgs_summary = '\n'.join(error_msgs)
+            raise ReadError(f"file could not be opened successfully:\n{error_msgs_summary}")
 
         elif ":" in mode:
             filemode, comptype = mode.split(":", 1)
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index 77ad8305c31..817e6f17997 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -2283,6 +2283,18 @@ class MiscTest(unittest.TestCase):
             'SubsequentHeaderError', 'ExFileObject', 'main'}
         support.check__all__(self, tarfile, not_exported=not_exported)
 
+    def test_useful_error_message_when_modules_missing(self):
+        fname = os.path.join(os.path.dirname(__file__), 'testtar.tar.xz')
+        with self.assertRaises(tarfile.ReadError) as excinfo:
+            error = tarfile.CompressionError('lzma module is not available'),
+            with unittest.mock.patch.object(tarfile.TarFile, 'xzopen', side_effect=error):
+                tarfile.open(fname)
+
+        self.assertIn(
+            "\n- method xz: CompressionError('lzma module is not available')\n",
+            str(excinfo.exception),
+        )
+
 
 class CommandLineTest(unittest.TestCase):
 
diff --git a/Lib/test/testtar.tar.xz b/Lib/test/testtar.tar.xz
new file mode 100644
index 00000000000..512fa149e6c
Binary files /dev/null and b/Lib/test/testtar.tar.xz differ
diff --git a/Misc/NEWS.d/next/Library/2021-03-13-14-02-07.bpo-8978.CRxG-O.rst b/Misc/NEWS.d/next/Library/2021-03-13-14-02-07.bpo-8978.CRxG-O.rst
new file mode 100644
index 00000000000..2da8874dc7f
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2021-03-13-14-02-07.bpo-8978.CRxG-O.rst
@@ -0,0 +1,2 @@
+Improve error message for :func:`tarfile.open` when :mod:`lzma` / :mod:`bz2`
+are unavailable.  Patch by Anthony Sottile.