Yes, you can get Mew in Pokémon Red/Blue Yellow! Just follow this simple guide.
First, get to this location (left picture) without beating the trainer with a single Slowpoke (right picture) on Route 25. Then save. Press Down and immediately press Start. If you did it correct you should see this:
Next, fly to Cerulean City. After this, nothing but the D-pad will work.
Walk to the aforementioned trainer and walk two or three spaces away, like this. If you step right in front of him, the game will freeze, like this:
Defeat his Slowpoke. The buttons should work again.
Fly to Lavender Town.
As you walk into Route 8, the menu will pop up automatically. Do not save the game. Instead, press B to close the menu (or choose Exit at the bottom).
You'll get into a battle with a Mew!
Catch it and save!
Here's its PokéDex entry.