About ⅔ or two thirds of the way, you can stop at Gusher, UT. This is a little over 1 hour southwest of Dutch John, or almost an hour east of Duchesne.
About ¾ or three fourths of the way, you can stop at Ballard, UT. This is about 1½ hours southwest of Dutch John, or about half an hour east of Duchesne.
If you drive straight through without stopping at all, the total distance is 102 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 1 hour, 59 minutes.