Because of my job I'm afraid of getting a papercut on my eye.
Do you think something bad will happen if you get a paper cut on your eyeball?
Yeah, I'm pretty sure you can go blind like that.
It's reasonable to be afraid of having sharp things near your eyes, as any damage to your eye can cause blindness. You should avoid putting anything sharp near your eyes.
It would be really bad for that to happen and at work there is constantly paper flying around. I tend to just close my eyes when it starts to happen and hope for the best.
Well, it's good that you're conscious of the danger but you should really just try to protect your eyes rather than closing them. Wear some protective glasses or goggles so you don't have to worry about having your eyes damaged. This is also helpful because now you won't have to close your eyes which can interfere with your job.