The opening narration confirms that the Dude is in fact, not a hero (emphasis mine):

[...] I only mention it because sometimes there's a man... I won't say a hero, 'cause, what's a hero? But sometimes, there's a man. And I'm talkin' about the Dude here. Sometimes, there's a man, well, he's the man for his time and place. He fits right in there. And that's the Dude, in Los Angeles. And even if he's a lazy man - and the Dude was most certainly that. Quite possibly the laziest in Los Angeles County, which would place him high in the runnin' for laziest worldwide. But sometimes there's a man, sometimes, there's a man. Aw. I lost my train of thought here. But... aw, hell. I've done introduced him enough.

Leon is a hitman, but he's also probably learning disabled. He rescued a girl from the mob.

The Dude is just a guy who likes bowling, but he's pretty cool and that's enough for me.

The Dude Abides.