Yes, believe it or not, it can be done without a GameShark or other cheating device.
Here's what you'll need to do:
* Restart your game and play until you get HM02 Fly from the house west of Celadon
* DO NOT fight the following trainers:
o The Youngster with a Slowpoke (Level 17) on the road to Bill's house (Route
25). He is the fourth trainer after you cross Nugget Bridge, standing directly above a
female trainer and facing north.
o The Gambler with two Poliwags and a Poliwhirl (all Level 22) between Lavender
Town and Saffron City (Route 8). He is standing just east of the entrance to Saffron
City, facing north toward the door of the Underground Path.
* Carefully follow the series of steps illustrated below.
Make sure you have a Pok√©mon that knows Fly. On Route 8, stand in front of the door of
the Underground Path. Save your game, since saving after this point will disrupt the
Walk down one step and immediately press Start. If you succeed, the menu will pop up
before the Gambler "sees" you. Go to the Pok√©mon list and fly to Cerulean City.
The Gambler will see you just before you fly away.
Your Start button will no longer function. Go north across Nugget Bridge and head east
until you find the trainer in this picture (he's the Youngster described above).
If you walk right up to the Youngster, the game will lock up, so you'll have to let him
see you and walk up to you. Defeat his Slowpoke (afterwards, your Start button will work
again), then fly to Lavender Town.
Head west into Route 8.
As soon as you enter the narrow path, the menu will pop up by itself. Press B to exit...
...and look who it is! Mew will be at Level 7 and its only attack will be Pound.
Once you've caught Mew, you can save your game without any ill-effects.
There are two alternative ways to enter Route 8 and trigger Mew after beating the
Youngster. If you have access to Saffron City, you can fly there and go through the
eastern city gate. Also, you can fly to Celadon City, head east to Route 7, enter the
Underground Path, and emerge back in front of the Gambler.
Finally, the Route 8 Gambler isn't the only trainer that can initiate the trick. The
Super Nerd just west of him can also do it, as can the second Fisherman you meet when
heading south from Lavender on Route 12. There are numerous others; basically, any
trainer who is scrolled onto the screen by the same step that puts you in their vision
range will work. However, remember that the final battle with Mew will only be triggered
by returning to whatever route that trainer is in, which is not necessarily Route 8.