I will tell a lil bit of my experience: staring at you is not a big problem yet. They might follow you. Someone used to come near to the store I used to work at and standing for hours, following me I end up complaining to the Security. Sometimes they follow you when you drive, they might put an eye right on the parking spot. I got physically attacked by a teen in the elevator, I had a big complain to a landlord and eventually to the police station. I mean, this country is still safe comparing to others, but you have always know how to deal with it and avoid the situations you dont wanna be at. For your safety, try to be at the decent areas.

And I guess you are new in Doha, my advice --- ignore, coz u wont be able to change it. Eventually you'll get used to this sometime soon and wont even notice that.