Diamond Only

Assuming you do not use any glitches and do not have the ability to get the encounters that require a GBA Pokémon cartridge inserted, the answer is 357. The Professor Oak challenge guide for Diamond can be found here, although I've made a shorter list of encounters below in case anything happens to the external guide.

With the ability to get the GBA dual-slot encounters, you can get an additional 38 Pokémon, bringing the total to 395.

The list of Pokémon you cannot get without trading or using dual-slot encounters, which equals 136 species, is as follows:

Two Starters (6) - 6 total: As you can only select one starter Pokémon, the other two starters, and their evolutions, are unavailable; this totals 6 Pokémon.
Trade Evolutions (16) - 22 total: Alakazam, Golem, Machamp, Gengar, Politoed, Slowking, Rhyperior, Kingdra, Scizor, Electivire, Magmortar, Porygon2, Porygon-Z, Dusknoir, Huntail, and Gorebyss evolve by trade. Without the ability to trade, they are unobtainable.
Spiritomb (1) - 23 total: Spiritomb requires the player to be able to connect with other players in the Underground. Without multiplayer access, it is unobtainable.
Version Exclusives (19) - 42 total: The following Pokémon cannot be obtained without the use of glitches or trading: Shieldon, Bastiodon, Misdreavus, Mismagius, Glameow, Purugly, Palkia, Slowpoke, Slowbro, Pinsir, Houndour, Houndoom, Stantler, Spheal, Sealeo, Walrein, Bagon, Shelgon, and Salamence.
Other Region Starters (27) - 69 total: All Kanto, Johto, and Hoenn starters must be transferred from Gen 3 or traded from HeartGold or SoulSilver.
Other Region Legendaries (17) - 86 total: All Legendaries from Kanto, Johto, and Hoenn must be transferred from Gen 3 or traded from Platinum, HeartGold, or SoulSilver.
Regigigas (1) - 87 total: Regigigas is only obtainable if the player has Regirock, Regice, and Registeel in their party. As those are locked to transfer or trade, Regigigas is thus unobtainable.
Miscellaneous (3) - 90 total: Tangela (and therefore Tangrowth) and Tropius must be transferred or traded from Platinum, HeartGold, or SoulSilver.
Dual Slot (37) - 127 total: The following Pokémon cannot be obtained without the use of the GBA dual-slot encounters: Caterpie, Metapod, Butterfree, Weedle, Kakuna, Beedrill, Ekans, Arbok, Sandshrew, Sandslash, Vulpix, Ninetales, Growlithe, Arcanine, Elekid, Electabuzz, Magby, Magmar, Pineco, Forretress, Gligar, Gliscor, Shuckle, Teddiursa, Ursaring, Lotad, Lombre, Ludicolo, Seedot, Nuzleaf, Shiftry, Sableye, Mawile, Zangoose, Seviper, Lunatone, and Solrock. (The 38th Pokémon is Gengar, which was mentioned separately.)
Mythical/Event (9) - 136 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 are difficult to come by.

Professor Oak Challenge Notes

Pre-Badge 1 (Roark) - 29 Total Pokémon

Badge 1 through Badge 2 (Gardenia) - 72 Total Pokémon

Badge 2 through Badge 3 (Maylene) - 110 Total Pokémon

Badges 3 through 6 (Byron) - 115 Total Pokémon

Badge 6 through 7 (Candice) - 123 Total Pokémon

Badge 7 through 8 (Volkner) - 129 Total Pokémon

Badge 8 through Elite Four - 130 Total Pokémon

Post-Elite Four - National Pokédex - 357 Total Pokémon

All National Pokédex Pokémon are only available after defeating the Elite Four.

If the player has access to dual-slot capabilities, the following Pokémon can also be obtained after obtaining the National Pokédex:

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How Many Pokémon Can You Get with Only Pokémon Diamond?