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.

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:

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How Many Pokémon Can You Get with Only Pokémon Diamond?