It's not a moment per se, but the 1998 Home Run Race was amazing. You got chills all over, goosebumps. It's easy now to, after knowing all we know now, to say "Ah, they're cheaters!", but no one can deny the importance that season had on baseball, and it was a great ride.
Here we see McGwire hitting #61, tying Roger Maris. Look at the crowd's reaction, I mean, everything. It's just awesome.