All of generation 1 & 2 can be acquired in either HeartGold or SoulSilver. However, with just HeartGold some will be unavailable.

 - The 2 johto and 2 kanto starters (and their evolutions) that you didn't choose will be unavailable. You need to trade to get the ones you didn't select.
 - Vulpix, Ninetales, Meowth, Persian, Kabuto, Kabutops, Ledyba, Ledian, Skarmory, and Delibird are unavailable, as they are SoulSilver exclusives.
 - Mew & Celebi are only available from giveaways, so it is unlikely you will be able to catch them.

This leaves a total of 24 unacquirable pokemon. You can consider your pokedex complete at **242**.

Source for version exclusive lists: https://bulbapedia.bulbagarden.net/wiki/Pok%C3%A9mon_HeartGold_and_SoulSilver_Versions#Game-exclusive_Pok.C3.A9mon