That's the premise of the critically acclaimed and Hugo award winning 2009 movie Moon, which stars Sam Rockwell as the moon base worker Sam.
You can watch Moon for free via Crackle on YouTube if you are willing to sit through a few ads.
Note that the video is rated "R", and so you'll need to sign in to your YouTube account to "prove" you are at least 18 years old.
[the rest of post has spoilers]
In the movie, Sam wakes up in the base's infirmary with no memory of the accident, and makes a startling discovery: he's not the Sam that was in the accident, but a replacement clone. The old Sam and the new Sam work together to figure out what's really going on before a "rescue" squad from the mining company arrives.
While the movie's director Duncan Jones has apparently thought a lot about how a Helium-3 mining facility on the Moon would operate, cloning is pretty much just a plot device. As Jones put it:
Cloning seemed to fit well into the embryonic story I was playing with of a man stuck in a moon base. I got excited thinking:And the movie raises what I think is an important philosophical question: if human cloning were cheap and easy, to what extent would cost-cutting businesses use clones if they were interchangeable and disposable parts of a machine? And how would people respond if that user were discovered?
“If you met you in person, would you like yourself?” I think it’s the most brutal, honest and human question there is…and that makes it perfect for sci-fi.”
Moon does not address the issue quite as as deeply or movingly as Kazuo Ishiguro's Never Let Me Go, but it still provides ample food for thought.
I think Rockwell did an excellent job playing two different versions of himself: one whose physical and mental health is breaking down, and a younger, healthier version who has not yet experienced the years of managing the base alone. It's thoughtful science fiction, rather than an action-packed blockbuster. I recommend it.
(If you prefer an ad-free version, you can watch for $0.99 through Amazon Instant)