Dogs will usually sit and stare at someone when they think that they will be getting something from the person such as food they're eating or maybe they're staring at you wanting you to take them for a walk.
I have a dog that will only sit and stare at me while I'm eating and when he does that he goes outside to his fenced in area of the yard while I eat.
It has corrected that behavior sometimes since now whenever I start to eat and the dogs stares at me I can get his leash and he'll go back to where he was before.
But sometimes he'll continue to stare and I immediately take him outside until I finish eating.
But then again a friend of mine has a dog that will only sit and stare at him when he wants to go outside to potty or just go for a walk so it can depend on the dog why they're staring at you.
Sometimes it could be they're wanting something you have, wanting you to do something for them or just wanting to engage you in a staring contest. lol