Quemado is on US-60, about 90 miles east of Show Low, Arizona, and 100 miles west of Socorro, New Mexico. There is no public transport. The nearest towns with services are Datil (25 miles east) and Reserve (50 miles south). A full tank of fuel is advisable before driving into this part of New Mexico: distances between services are long and the terrain unforgiving if you run dry.
Socorro (on the Rio Grande, about 90 miles east) is the nearest town with motels, restaurants, and a full range of services, and makes a more comfortable base if you want to visit the VLA without staying in Quemado itself.