Is it possible for someone to be divorced and not know it?

Sure. It's possible to divorce on the grounds of abandonment, so it's entirely possible that a person could divorce on this grounds without the other person being aware that it's been done. My ex husband divorced his first wife this way. She left and he had no idea where she went. He eventually left the state and had no way of contacting her. He filed for divorce on the grounds of abandonment. He had to provide the court with all the addresses and numbers he'd had for her and publish a notice in newspapers in areas where he'd known her to live, giving her 30 days to contest the divorce. Once the ads ran for 30 days and her 30 days to contest elapsed, the court granted him a final judgment of divorce. For all I know, she still doesn't know she's divorced.

No - sorry - but who ever is telling you that must be really convincing.

How are you supposed to get divorced without signing papers?

I mean, are you even serious asking this?