Why aren't 4096-bit RSA PGP key blocks $\lceil\frac{4096}{6}\rceil=683$ base64 characters long? If a 4096-bit RSA PGP key is considered sufficiently long for most purposes, why are so many key blocks around 3000 base64 characters in length? Why aren't $\lceil\frac{4096}{6}\rceil=683$  base64 characters enough?
Base64 uses $64=2^6$ characters, and hence can encode at most 6 bits per character.  Hence, the correct calculation would be at least $\frac{4096}{6} = 682 \frac{2}{3} $ characters (or 683, we really can't have fractional characters)

As far as why PGP keys are longer than that, well, that's because they're carrying around a lot more information than just the raw RSA key.