catch  (Exception e) {
   String msg = e.getMessage();
   #sql {insert into t values (:msg)};
}

SQL> create or replace and compile java source named "Test" as
   2    import  java.sql.SQLException;
   3    public   class  Test {
   4      public   static   void  foo()  throws  SQLException {
   5        try  {
   6          throw   new  SQLException ("My exception");
   7       }  catch  (Exception e) {
   8         String message = e.getMessage();
   9         #sql {insert into TMP_ERROR(message) values (:message)};
  10       }
  11     }
  12   }
  13   /

Java created.

SQL> create table TMP_ERROR (message varchar2( 4000 ));

Table created.

SQL> create or replace procedure test_foo as language java name 'Test.foo()';
   2   /

Procedure created.

SQL> exec test_foo;

PL/SQL procedure successfully completed.

SQL> col message format a100
SQL> select * from TMP_ERROR;

MESSAGE
-----------------------------------------------------------------------------
My exception