Re: '300 years ago' , in this film's context, was the 1950s!
It is possible that it was the 1950s in which Khan and his Augments were bred. If so, that would place Khan in his forties when he fought the Eugenics Wars. Coincidently enough, Ricardo Montalban was in his forties when he was in Space Seed. Cumberbatch is only 36, but presumably genetic engineering can make people look younger than they are.
"Internet message boards aren't as funny today as they were ten years ago. I've stopped reading new posts." -The Simpsons 20th anniversary special.