Question

How do I get Mew to be an entry in the Pokedex in Pokemon Red? Is it possible to catch Mew in Pokemon Red?

Answer 1

Not as a sprite, or a legendary (like mewtwo) if that's what you are asking, but you can still get him pretty easely. I still have my old cartridge of Yellow from when I was a kid, and the Red one I stole from my "best friend" of then. (best Italian behavior right here)
I still have the untouched completed game in Yellow from more than a decade ago, and the pretty recently refreshed Red save file.
In both I have Mew, so yes, you can and it is by no means necessary to get an event to make him appear in generation 1, but you WILL need to perform a "glitch" so to speak.
I used what here is described as the 2nd way to catch him (the one with Abra), so I can guarantee they work.
I don't think I have to copy and paste the sentences from the site if I can provide a link, correct? If I do, just tell me and I'll edit the message.
http://www.wikihow.com/Find-Mew-in-Pokemon-Red/Blue

Answer 2

It's impossible to catch Mew in any game without an event.

Since events for DS games, let alone Gameboy games, are long over, there is no way to get Mew legitimately.

You can use action replay if you really want to have Mew in Red, either exploit a glitch or hack.