No, you could set it today, as long as you still hit the major beats of the story. In fact, a contemporary setting would work better today than at any time over the last 50 years. One of the biggest criticisms of Stan and Jack's origin story was 'How on Earth do you sneak your girl friend and her kid brother on to a space flight'? But that was when only government agencies had the will or the finance to send rockets up. In today's climate we have private individuals - Elon Musk and SpaceX for example - getting in on the act. If Reed was a big noise in such a company it would surely be much easier for them to circumvent any security.

As long as you stick to the major points (The ship should launch in the near future, something happens which necessitates they launch NOW, but the can't get the clearances or it's too dangerous, Reed & co take the ship up, cosmic rays, crash landing back on earth), then the fans would be happy.