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How Many Pokémon Can You Get with Only Pokémon HeartGold?

Assuming you do not use any glitches and do not have access to the Pokéwalker, the answer is 373. The Professor Oak challenge guide for Gold can be found here, although I've made a shorter list of encounters below in case anything happens to the external guide.

With Pokéwalker access, you can get an additional 21 Pokémon, bringing the total to 394. The guide for using Pokéwalker is located here.

The list of Pokémon you cannot get without trading or using the Pokéwalker, which equals 120 species, is as follows:

Two Johto Starters (6) - 6 total
As you can only select one Johto starter Pokémon, the other two starters, and their evolutions, are unavailable; this totals 6 Pokémon.
Two Kanto Starters (6) - 12 total
As you can only select one Kanto starter Pokémon, the other two starters, and their evolutions, are unavailable.
Two Hoenn Starters (6) - 18 total
As you can only select one Hoenn starter Pokémon, the other two starters, and their evolutions, are unavailable.
Trade Evolutions (16) - 34 total
Alakazam, Machamp, Golem, Gengar, Politoed, Slowking, Steelix, Scizor, Kingdra, Porygon2, Dusknoir, Huntail, Gorebyss, Rhyperior, Electivire, Magmortar, and Porygon-Z evolve by trade. Without the ability to trade, they are unobtainable.
Location-based Evolutions (4) - 38 total
Magnezone, Probopass, Leafeon, and Glaceon all evolve by level-up in a specific location (a magnetized area, a Mossy Rock, or an Icy Rock). These locations are not in HeartGold and SoulSilver, rendering these Pokémon unobtainable.
Version Exclusives (19) - 57 total
The following Pokémon cannot be obtained without the use of glitches or trading: Vulpix, Ninetales, Meowth, Persian, Kabuto, Kabutops, Ledyba, Ledian, Teddiursa, Ursaring, Delibird, Skarmory, Mawile, Gulpin, Swalot, Lileep, Cradily, Latios, and Groudon.
Pokéwalker Pokémon (21) - 78 total
Skitty, Delcatty, Carvanha, Sharpedo, Wailmer, Wailord, Feebas, Milotic, Castform, Kecleon, Tropius, Snorunt, Glalie, Froslass, Shellos, Gastrodon, Spiritomb, Finneon, Lumineon, Snover, and Abomasnow must be obtained from the Pokéwalker, or by trading from Diamond, Pearl, or Platinum or transferring from Ruby, Sapphire, or Emerald.
Sinnoh Pokémon (20) - 98 total
Turtwig, Grotle, Torterra, Chimchar, Monferno, Infernape, Piplup, Prinplup, Empoleon, Cranidos, Rampardos, Shieldon, Bastiodon, Drifloon, Drifblim, Glameow, Purugly, Stunky, Skuntank, and Rotom must be traded from Diamond, Pearl, or Platinum.
Legendary Pokémon (13) - 111 total
Regirock, Regice, Registeel, Rayquaza, Uxie, Mesprit, Azelf, Dialga, Palkia, Heatran, Regigigas, Giratina, and Cresselia must be obtained by trading from Diamond, Pearl, or Platinum, or transferring from Ruby, Sapphire, or Emerald.
Mythical/Event (9) - 120 total
Outside of glitches, Mew, Celebi, Jirachi, Deoxys, Phione, Manaphy, Darkrai, Shaymin, and Arceus were only obtainable via events that are no longer active, or by transferring from another game.

Professor Oak Challenge Notes

Pre-Badge 1 (Falkner) - 39 Total Pokémon (100 with Pokéwalker)

By speaking to Primo in Violet City, the player can obtain a Slugma egg in exchange for a password. If you choose to get this egg, you can get Slugma and Magcargo for a total of 41 Pokémon.

With access to the Pokéwalker, the player can obtain the following Pokémon. Since Slowpoke can be obtained here, so can Headbutt encounters; you will have to train Slowpoke to level 25. The total with Pokéwalker access is 102.

Badge 1 through Badge 2 (Bugsy) - 66 Total Pokémon (108 with Pokéwalker)

To use Headbutt here before being able to use Cut, you will need to use Slowpoke, who learns Headbutt at level 25.

Badge 2 through Badge 3 (Whitney) - 97 Total Pokémon (126 with Pokéwalker)

Badge 3 through Badge 4 (Morty) - 137 Total Pokémon (152 with Pokéwalker)

Badge 4 through Badge 5 (Pryce) - 174 Total Pokémon (180 with Pokéwalker)

Badges 5 through Badge 7 (Chuck) - 176 Total Pokémon (182 with Pokéwalker)

Badge 7 through Badge 8 (Clair) - 188 Total Pokémon (192 with Pokéwalker)

Badge 8 through Elite Four - 195 Total Pokémon (197 with Pokéwalker)

Elite Four through Badge 9 (Misty) - 348 Total Pokémon (374 with Pokéwalker)

For the Safari Zone, the Professor Oak guide linked at the top of the page outlines a strategy of setting up areas and objects in the Safari Zone to get all the Pokémon there as quickly as possible. The suggestions given there are recreated below. Pokémon that can be obtained outside the Safari Zone have their alternate location mentioned.

First, set the six areas of the Safari Zone to Rocky Beach, Wetland, Meadow, Forest, Peak, and Savannah.

In Rocky Beach, you can get the following Pokémon immediately:

Wetland:

Meadow:

Peak:

Savannah:

After 20 total days:

Savannah:

Forest:

After 30 total days:

Savannah:

Wasteland:

Meadow:

After 40 total days:

Peak:

Desert:

After 60 total days:

Desert:

Wasteland:

Swamp:

Mountain:

After 70 total days:

Marshland:

Wetland:

Meadow:

Forest:

After 80 total days:

Marshland:

After 100 total days:

Rocky Beach:

Evolution stones can all be purchased at the Pokéathlon. You can also evolve Togetic, Misdreavus, and Murkrow, for a total of 258 before starting the rest of the section.

The rest of the Pokémon in this section can be obtained normally in other locations.

With access to the Pokéwalker, the player can obtain the following Pokémon, bringing the total to 374.

Badge 9 through Badge 16 - 349 Total Pokémon (375 with Pokéwalker)

Badge 16 through defeating Red - 366 Total Pokémon (387 with Pokéwalker)

After defeating Red - 373 Total Pokémon (394 with Pokéwalker)