When I first played through Red, I didn't care too much about completing my Pokedex, so I just picked up one Eevee and evolved it. Unfortunately, all of my friends evolved theirs into the same form, so trading wouldn't have been too useful. I suppose another reason why I didn't want to trade too much anyway was because almost everyone I knew had Red, and the only person who had Blue had a game that was so badly glitched that none of the rest of us wanted to get near it.
However, I became more interested in completing Pokedexes after finishing the one on my Gold. After Gold died, I finally decided to go back and complete my Red Pokedex, so I grabbed the Eevee from Silver or Crystal and bred it to get an extra Eevee, then sent both of those over to Red to evolve. So, strictly speaking, I never obtained three separate Eevees in Gen 1 itself, but I did manage to acquire enough of them to complete my Pokedex eventually.