Watch for oncoming motorcycles, especially in wash gullies where the sound of their sputtering motors is muffled. Any chance of timely organized rescue is farfetched. It's an hour's drive to Green River from either trailhead. If you are stranded midroute, it's a day's walk just to reach one of the trailheads. Passersby are unlikely.
Green River has a medical clinic.
The closest hospital is in Price, several hours away. From Green River drive 11 miles west on Interstate 70 and take Exit 147 for Utah State Highway 24 and Hanksville. Travel 25 miles south on UT 24. At milepost 137 turn right for Goblin Valley and Temple Mountain, and drive 5 miles on the paved road to the Goblin Valley turnoff. Where the road breaches the Reef, pull off to the right and park a vehicle in the dirt clearing on the south side of Temple Mountain.