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How Many Pokémon Can You Get with Only Pokémon Red?

Assuming you do not use any glitches, the answer is 124. The Professor Oak challenge guide for Red can be found here, although I've made a shorter list of encounters below in case anything happens to the external guide.

The list of Pokémon you cannot get without trading, which equals 27 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 (4) - 10 total
Golem, Machamp, Alakazam, and Gengar evolve by trade. Without the ability to trade, they are unobtainable.
Two Eevee evolutions (2) - 12 total
As you only receive one Eevee, you can only receive one of its evolutions.
One Fossil Pokémon (2) - 14 total
As you can only obtain one Fossil in Mt. Moon, you miss out on the other Fossil Pokémon (Omanyte or Kabuto) and its evolution.
One of Hitmonlee/Hitmonchan (1) - 15 total
As you can only obtain one ot Hitmonlee or Hitmonchan, you miss out on the other.
Version Exclusives (11) - 26 total
The following Pokémon cannot be obtained without the use of glitches or trading: Sandshrew, Sandslash, Vulpix, Ninetales, Meowth, Persian, Bellsprout, Weepinbell, Victreebel, Magmar, and Pinsir.
Mythical/Event (1) - 27 total
Outside of glitches, Mew was only obtainable via events that are no longer active.

Professor Oak Challenge Notes

Pre-Badge 1 (Brock) - 21 Total Pokémon

Badge 1 through Badge 2 (Misty) - 47 Total Pokémon

Badge 2 through Badge 3 (Koga) - 105 Total Pokémon

Badge 3 through Badge 4 (Erika) - 121 Total Pokémon

Badges 4 through 8 - 122 Total Pokémon

Badge 8 through Elite Four - 123 Total Pokémon

Post-Elite Four - 124 Total Pokémon