diff --git a/synapse/storage/engines/postgres.py b/synapse/storage/engines/postgres.py
index cb79e3b15602..b350f57ccb4a 100644
--- a/synapse/storage/engines/postgres.py
+++ b/synapse/storage/engines/postgres.py
@@ -22,7 +22,7 @@
     IncorrectDatabaseSetup,
     IsolationLevel,
 )
-from synapse.storage.types import Cursor, DBAPI2Module
+from synapse.storage.types import Cursor
 
 if TYPE_CHECKING:
     from synapse.storage.database import LoggingDatabaseConnection
@@ -35,9 +35,7 @@ class PostgresEngine(
     BaseDatabaseEngine[psycopg2.extensions.connection, psycopg2.extensions.cursor]
 ):
     def __init__(self, database_config: Mapping[str, Any]):
-        # Cast: mypy 1.0.0 doesn't seem to think that the module implements the protocol.
-        # AFAICS this is a false positive.
-        super().__init__(cast(DBAPI2Module, psycopg2), database_config)
+        super().__init__(psycopg2, database_config)
         psycopg2.extensions.register_type(psycopg2.extensions.UNICODE)
 
         # Disables passing `bytes` to txn.execute, c.f. #6186. If you do
diff --git a/synapse/storage/engines/sqlite.py b/synapse/storage/engines/sqlite.py
index 6df470dc41be..28751e89a5a5 100644
--- a/synapse/storage/engines/sqlite.py
+++ b/synapse/storage/engines/sqlite.py
@@ -15,10 +15,10 @@
 import sqlite3
 import struct
 import threading
-from typing import TYPE_CHECKING, Any, List, Mapping, Optional, cast
+from typing import TYPE_CHECKING, Any, List, Mapping, Optional
 
 from synapse.storage.engines import BaseDatabaseEngine
-from synapse.storage.types import Cursor, DBAPI2Module
+from synapse.storage.types import Cursor
 
 if TYPE_CHECKING:
     from synapse.storage.database import LoggingDatabaseConnection
@@ -26,9 +26,7 @@
 
 class Sqlite3Engine(BaseDatabaseEngine[sqlite3.Connection, sqlite3.Cursor]):
     def __init__(self, database_config: Mapping[str, Any]):
-        # Cast: mypy 1.0.0 doesn't seem to think that the module implements the protocol.
-        # AFAICS this is a false positive.
-        super().__init__(cast(DBAPI2Module, sqlite3), database_config)
+        super().__init__(sqlite3, database_config)
 
         database = database_config.get("args", {}).get("database")
         self._is_in_memory = database in (
diff --git a/synapse/storage/types.py b/synapse/storage/types.py
index 0031df1e0649..77adaa485775 100644
--- a/synapse/storage/types.py
+++ b/synapse/storage/types.py
@@ -12,7 +12,18 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 from types import TracebackType
-from typing import Any, Iterator, List, Mapping, Optional, Sequence, Tuple, Type, Union
+from typing import (
+    Any,
+    Iterator,
+    List,
+    Mapping,
+    Optional,
+    Sequence,
+    Tuple,
+    Type,
+    Union,
+    Callable,
+)
 
 from typing_extensions import Protocol
 
@@ -112,15 +123,35 @@ class DBAPI2Module(Protocol):
     #   extends from this hierarchy. See
     #     https://docs.python.org/3/library/sqlite3.html?highlight=sqlite3#exceptions
     #     https://www.postgresql.org/docs/current/errcodes-appendix.html#ERRCODES-TABLE
-    Warning: Type[Exception]
-    Error: Type[Exception]
+    #
+    # Note: rather than
+    #     x: T
+    # we write
+    #     @property
+    #     def x(self) -> T: ...
+    # which expresses that the protocol attribute `x` is read-only. The mypy docs
+    #     https://mypy.readthedocs.io/en/latest/common_issues.html#covariant-subtyping-of-mutable-protocol-members-is-rejected
+    # explain why this is necessary for safety. TL;DR: we shouldn't be able to write
+    # to `x`, only read from it. See also https://github.com/python/mypy/issues/6002 .
+    @property
+    def Warning(self) -> Type[Exception]:
+        ...
+
+    @property
+    def Error(self) -> Type[Exception]:
+        ...
 
     # Errors are divided into `InterfaceError`s (something went wrong in the database
     # driver) and `DatabaseError`s (something went wrong in the database). These are
     # both subclasses of `Error`, but we can't currently express this in type
     # annotations due to https://github.com/python/mypy/issues/8397
-    InterfaceError: Type[Exception]
-    DatabaseError: Type[Exception]
+    @property
+    def InterfaceError(self) -> Type[Exception]:
+        ...
+
+    @property
+    def DatabaseError(self) -> Type[Exception]:
+        ...
 
     # Everything below is a subclass of `DatabaseError`.
 
@@ -128,7 +159,9 @@ class DBAPI2Module(Protocol):
     # - An integer was too big for its data type.
     # - An invalid date time was provided.
     # - A string contained a null code point.
-    DataError: Type[Exception]
+    @property
+    def DataError(self) -> Type[Exception]:
+        ...
 
     # Roughly: something went wrong in the database, but it's not within the application
     # programmer's control. Examples:
@@ -138,28 +171,45 @@ class DBAPI2Module(Protocol):
     # - A serialisation failure occurred.
     # - The database ran out of resources, such as storage, memory, connections, etc.
     # - The database encountered an error from the operating system.
-    OperationalError: Type[Exception]
+    @property
+    def OperationalError(self) -> Type[Exception]:
+        ...
 
     # Roughly: we've given the database data which breaks a rule we asked it to enforce.
     # Examples:
     # - Stop, criminal scum! You violated the foreign key constraint
     # - Also check constraints, non-null constraints, etc.
-    IntegrityError: Type[Exception]
+    @property
+    def IntegrityError(self) -> Type[Exception]:
+        ...
 
     # Roughly: something went wrong within the database server itself.
-    InternalError: Type[Exception]
+    @property
+    def InternalError(self) -> Type[Exception]:
+        ...
 
     # Roughly: the application did something silly that needs to be fixed. Examples:
     # - We don't have permissions to do something.
     # - We tried to create a table with duplicate column names.
     # - We tried to use a reserved name.
     # - We referred to a column that doesn't exist.
-    ProgrammingError: Type[Exception]
+    @property
+    def ProgrammingError(self) -> Type[Exception]:
+        ...
 
     # Roughly: we've tried to do something that this database doesn't support.
-    NotSupportedError: Type[Exception]
+    @property
+    def NotSupportedError(self) -> Type[Exception]:
+        ...
 
-    def connect(self, **parameters: object) -> Connection:
+    # We originally wrote
+    # def connect(self, *args, **kwargs) -> Connection: ...
+    # But mypy doesn't seem to like that because sqlite3.connect takes a mandatory
+    # positional argument. We can't make that part of the signature though, because
+    # psycopg2.connect doesn't have a mandatory positional argument. Instead, we use
+    # the following slightly unusual workaround.
+    @property
+    def connect(self) -> Callable[..., Connection]:
         ...