Let me ask you one question, were you able to login to your orcl instance using sys and system as the user names through sqlplus. On such case your database is in a good shape. You don't have to worry much about the database ...
What is the gurentee that blowing every thing off and starting things from the scratch would solve your problem.
When you say that you tried to connect as system, what was you using, sqlplus/OEM?
Comming back to square one, do you have the OEM repository configured on any oracle instace?
Could you be more clear where you get the failure? I also need to know your working OS(es)? i.e. what is the OS the server is on and what is the OS the OEM is on and etc ...
Looking forward to hear from you ...
Life is a journey, not a destination!