My brother and sister-in-law were staying with my mother when she died. I inherited her house and want them out because I intend to rent it out. I first gave them 3 months (verbally); then a written 30-day notice to vacate; and now a final notice to vacate in 2 weeks. I'm tired of them dragging their feet. No doubt they will dash out of the house at the very last minute, leaving some of their stuff behind, which they will expect me to keep until they can retrieve it, at their convenience.

So what am I allowed to do with their possessions? Can I give them written notice now that stuff left behind will be considered "abandoned" so I can dispose of it as I see fit? Or do I have to keep their possessions for some specified time? How long? Can I move it out of my house and put it in storage and make them pay?

I'm in Maryland, if that makes a difference.

You are going to have to go to your local court or magistrate and show them you gave them the two written orders and get a court order. Then when the court order date comes if they will not leave the sheriff will come and make them leave.
You then need to keep the things they leave behind for whatever amount of time your state law requires before doing anything with it. Ask when you get the court order. Generally it is a minimum 30 days.
You can put it in storage probably after the 30 days but you would have to make the initial payment if they won't.