Sunken eyes do not necessarily or exclusively mean you're close to death. The correlation has to do with the fact that when someone IS close to death, various physical and physiological changes occur to the body, only one of which is the appearance of "sunken in" eyes.
A number of different diseases can also cause this, so can lack of sleep, so can various vitamin deficiencies, stress,...
In fact, you can show one or several of the same "symptoms" as someone who is dying, but not be dying.
If you are concerned about it, see your doctor, - especially if you have any other physical problems too. Other than that, take a closer look at how you're sleeping and what other stressors are in your life.