There is no NPS-managed lodging within Great Basin National Park. You’ll need to look outside the park’s boundaries for brick-and-mortar places at which to lay your head at night.
The nearest town with consistent lodging is Ely (Eee-lee) Nevada, about an hour’s drive south and east of the park. Baker Nevada (pop 21 as of 2022) offers some lodging (including Airbnbs) that is usually snapped up quickly.
A few miles across the Nevada / Utah state line along Utah State Route 21 is the exit from Garrison, Utah, down seven miles (11.3 km) of graded dirt and gravel road back into Nevada to Hidden Canyon Retreat. Located within a hidden canyon, this gem offers three room types which include AC, fridge, small microwave, bagel toaster, small toaster oven, coffee maker, and kitchen-style sink in addition to the regular hotel/motel amenity offerings. Cell service is available around the area and the Retreat offers internet, although it is so weak that you really can only use it for email and texting, at best. Hotel-style rooms for 2024 run approximately $179 - $219 standard rate, not including taxes, to simpler camping cabins going for $75 per night.
Where ever you choose to stay, it's a good idea to make your lodging reservations months ahead of your actual stay near Great Basin. It’s also a good idea to not rely on your GPS out there, and especially not at night.