Enable archivelog

As the ORA-19815 message indicates, your 3GB recovery destination is full; the database is unable to archive any more redo log files. It has reached a point where it needs to switch a full redo log file out, but there are none available for re-use because they can't be archived. I imagine this is why it wouldn't shut down cleanly, and possibly why you wanted to shut down in the first place as you wouldn't have been able to do much with the database in that state.

The short-term fix is to increase the size of the recovery area, assuming you have enough disk space, e.g.:

startup nomount
alter system set db_recovery_file_dest_size=4G scope=both;
alter database mount;
alter database open;

But you then need to work out how to manage your archives, and set the recovery area size to a sensible value for your backup and retention needs. This may be as simple as scheduling a job to remove obsolete backups, e.g. from the Enterprise Manager console's Availability tab, under Manage Current Backup; but you need to determine the best course for you (and not blindly follow advice from some random guy on the Internet who knows nothing about your requirements, and not that much about EM/RMAN either).

You need to restore/recover your database. If you can't then yes the code is buried in the system table spaces. But it won't be easy looking through the data file with a hex editor trying to find the code. You should either try to restore the database or just rewrite the code. Do you have a backup of your database? Do you have all of the data files, control files, online redo files? If you don't have a backup and your database is in an inconsistent state, then you can probably alter database recover until cancel and then point Oracle to your online redo logs to see if it can find what it needs.

Otherwise you might be out of luck. What is in your alert log for your database?