If you're looking for a meeting place or a good stop around ½ or halfway between Ashburn and Elkton, you can try Baltimore, MD, which is about 1 hour east of Ashburn, or about 1 hour southwest of Elkton.
Another possible place to stop near the midpoint is Ellicott City.
About ¼ or a fourth of the way, you can stop at Silver Spring, MD. This is about half an hour east of Ashburn.
About ⅓ or a third of the way, you can stop at College Park, MD. This is almost an hour east of Ashburn.
About ⅔ or two thirds of the way, you can stop at Essex, MD. This is a little over 1 hour east of Ashburn, or almost an hour southwest of Elkton.
About ¾ or three fourths of the way, you can stop at Edgewood, MD. This is about 1½ hours east of Ashburn, or about half an hour southwest of Elkton.
If you drive straight through without stopping at all, the total distance is 119 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 2 hours, 4 minutes.