I had a bunch of weird experiences. Like a man walking up to me on the sidewalk, starting to talk like we knew each other for years. He had to ask my name though, so he didn't mistake me for someone else. Finally he even pulled out a violin and started playing some classical music. Quite good I'd say, people going past us actually stopped and listened for a while.
But the weird experience that tops it all happened to me when I was a 14.
I was waiting at the central station in my hometown when I realized that an elder woman was watching me. I didn't give it a second thought though. Then my friend called and we talked briefly. As I hung up and turned around, the woman stood right in front of me. Suddenly she starts babbeling some completely random and and incomprehensible stuff, half German half English (I'm Austrian). I can't remember most of it, just her extremely insulting tone, like she was upbraiding me. But she kept her voice down, so noone really took notice of us.
Being the shy and introverted girl that I was, I didn't know what to do or say, so I just walked away. And then she actually followed me through the whole station. Walking faster, stopping abruptly and looking at her, nothing helped. I finally opened my mouth and said "Could you please leave me alone?" in the nicest way I could manage. Then again, I tried to get away from her as I felt a sort of kick in my heel. I turned around, but she looked back like nothing happened. I wasn't sure, if I imagined that feeling or not, so I just continued walking. Then I felt it again and almost freaked out. At that point, however, my train finally arrived in the station and I quickly ran to the doors.
She kept staring at me from the outside until the train left.