Holy. Crap. This mirrors my situation; but I'm the girl in the story and we have gotten to the hanging-out-outside-of-work-stage by now. So very close to my own situation! From the age difference to the introverted personality to the trust issues and the threat of gossip. Amazing.

Assuming that the lady in your situation and I are alike even a little bit mentally, I'd say the fact that she smiles at you is a great indicator that she may like to get to know you. At the very least, she acknowledges your existence and flashes a hint of happiness at you rather than avoiding eye contact or frowning at you. When the guy in my situation and I met at work, I smiled at him every chance I got, hoping he'd pick up on it and eventually talk to me. He smiled back, and that started the formation of a friendly bond between us.

Start off slow. Trust issues.... indeed they could have root in something "bad to worse" occurring in her past; that rings true for as well. Just be "friendly" and don't freak her out by moving forward too fast. Strike up conversation about work, move in to nerdy stuff or something she may find interesting. Move slow, move soft. Maybe she could use a friend, and maybe she could use something more. If you hit it off, meeting outside of work on your own time should be an option in the future.

Best of luck.