Does special relativity affect biological processes, such as aging?
The biological process, as such, is not affected. However, the way other people perceive your aging is greatly affected.
So let us say that you are 20 years old and you climb into a rocket, going 99.5% of the speed of light for five years (round trip) of the time you measure. You land back on Earth with the body of a 25-year-old just the same as if you had spent your five years on Earth. But since you were in the rocket, fifty years passed by according to your friends who stayed behind. If any of them survived, they will see you aged only five years in their past fifty years. But according to your own body, you aged five years in five years.
It works for microscopic particles too. A group of identical particles might have a half-life of 10 microseconds as measured by their own time. A scientist will duly measure the half-life as 10 microseconds if the particles are just sitting there in front of him. But if instead the particles are accelerated to 99.5% of the speed of light, the scientist sees the half-life grow to 100 microseconds on his watch even though it's still only 10 microseconds according to the particles.
Special relativistic effects change not the process within the particles or your body, but an observer's timing of that process.