[Odessa Medical Center Hospital] - The Portal to Texas History. Health (5 days ago) On the back there is print on the top left that reads: "Odessa's 408-bed Medical Center Center Hospital is a county-owned institution staffed by all of the city's 69 Physicians ans Surgeons. It's a medical center for a wide area of West Texas, and is one of the most modern and finest equipped hospitals in the ....