Bilkent University is designed in such a manner that you will find everything you would need all on campus or in close proximity.
On campus, you will have access to a (almost entirely) free health center that functions 24/7. Here you will be able to visit doctors, take x-rays, take blood tests, and much more. You will also have access to a pharmacy which gives 80% discounts to students that bring prescriptions from the Bilkent health center.
In terms of sports, the Main Campus has one big sports center where you can play sports such as squash, badminton, basketball, and volleyball. You can also join special classes like yoga. Moreover, there is a gym and an indoor swimming pool. There is also a second, but smaller, sports center. For more information, you can read here.
You can buy textbooks from the Meteksan Bookstore and shop for groceries from the small Meteksan Supermarket. Otherwise, next to the university you will find a hypermarket called Real in Bilkent Center. For more information about Real and Bilkent, check this blog post by a Bilkent student.
Lastly, there are dry-cleaning services, hair and beauty salons for females, and a barber shop.