What is the meaning of the phrase 'sometimes the bar eats you' in The Big Lebowski?

Bar is an antiquated variation of bear. Frontiersman Daniel Boone, for instance, famously carved on a tree once, after killing a bear:

D. Boon Cilled a Bar on [this] tree in the year 1760.

It makes sense that The Stranger would use such a colloquialism, as he's a sort of personification of the Old West (which is what he means when he says it's "far from" being Eastern). The saying is attributed to baseball pitcher Preacher Roe (though it was possibly older):

After being taken out of a game in the second inning, Roe commented that, "Sometimes you eat the bear and sometimes the bear eats you."

The phrase's general meaning is: You win some, you lose some; There are good days and bad. The Stranger says it to the dispirited Dude to try and make him feel a little better.

As noted by others, he's not saying bar, as in a place to get drinks, he's saying bear as in the animal.

The US is full of regional dialects and accents, and the narrator is supposed to be some kind of cowboy type. He speaks with a fairly thick Texas drawl, and the word bear just happens to sound like bar.

For example, take the Ka-Bar style of knife. The reason it's called a Ka-Bar? The company that makes this style of knife once received a letter from a fur trapper with testimony of how he used their knife to defend himself against a wounded bear that attacked him after his rifle jammed. The letter wasn't terribly legible, and of the fragments they could make out, "ka bar" was fairly easily understood to mean "kill a bear".

The bear is a metaphor for life, and the phrase thus generally means, "Sometimes you win at life, and sometimes you lose."

Y'all are thinking too deep on this one. The meaning of the saying does not change when substituting 'bear' for 'bar'. However considering how much drinking and drug use is present in the movie and the fact that they are sitting at the BAR while having this conversation, it only adds more flavor to the punchline.

And for those who don't know what 'the bar eats you' means, you must have never gotten thoroughly wasted (and good for you that you haven't, I do not recommend it; in my experiences the bar always wins. Having been sober for nearly 20 years I still sometimes have eerie flashbacks and dark memories).