You're right, there was no EarthBound for the NES. However, Nintendo had translated the Japanese game Mother, and was scheduled for release in the US as EarthBound in the Fall of 1991. Unfortunately, it was never released. Fortunately, someone acquired the prototype, and another person dumped it. Unfortunately, if you aren't the prototype owner, the only way to play this game is through emulation. Fortunately, the ROM is pretty common; anyone using search engines is bound (pun intended) to find it.
If you didn't already know, an EarthBound was released on the Super NES, but it is called Mother 2 in Japan. Its predecessor, Mother, was released for the Famicom. Many EB fans refer to the unreleased translated NES game as "EarthBound Zero."
Some kid in the Twinkle Elementary School says: "Have you played Super Mario Bros. 7? I'm still playing Super Mario Bros. 3. It's been quite a challenge for me."
In the Japanese game (Mother), the same kid originally said: "Have you played Dragon Quest 4? I'm still having a hard time with 3." (Dragon Quest is the Japanese name for Dragon Warrior.)
Thanks to Tomato of Starmen.net
for his help with the translation of the sentence from Mother.