No, Bill Clinton Can't Run for the French Presidency

No, this man cannot become
the next president of France.

There appears to be a persistent belief that if you live in any of the US states acquired in the Louisiana Purchase, you can quickly acquire French citizenship. This was sparked, in part, by none other than Bill Clinton claiming that he could become President of France. He said:

And because I was born in Arkansas, which is part of the Louisiana Purchase, any person anywhere in the world that was born in a place that ever was part of the French empire, if you move to — if you live in France for six months and speak French, you can run for president.

Bill Clinton said this in 2012, but was unaware that this interesting quirk of French law was eliminated in 2006.
After you begin to understand how immigration works, the idea that a resident of Arkansas could quickly become a French citizen seems ludicrous. However, once you learn more and find out how strange some laws are, maybe it doesn't seem that ludicrous. For example, are you of Italian descent? If had an Italian ancestor since 1861 (more or less), you might be able to claim Italian citizenship so long as no one along your chain of ancestors renounced their Italian citizenship. (If you're curious to know if you're eligible, this Web site questionnaire lets you check your eligibility for Italian citizenship by ancestry)
Yeah, laws are strange and it's hard to follow them. What's worse, researching on the Web leads you to articles like this. That article claims that you might have ways of becoming a French citizen (or shortening your time to become one) if you were a Louisiana resident (no), were married to a French citizen for at least 12 months (no longer true), or have had a child with a French citizen (no). I fall into the last two categories and I guarantee that they're not true, but the person doing the research obviously found some information because it's all not entirely far from the truth and some of that information was true shortly before that article was published.
If you're married to a French national, you can apply for citizenship after four years (three years if you're living in France). Or:

• If you have a child with French citizenship
• and you can show that you're been proving support for the child for at least two years
• and you swear that you're not polygamous (this being aimed Africans and/or Muslims)

Then you can apply for residency in France, but normal citizenship rules apply. That's how I'm currently updating my visa, by the way, because the civil servants assured me that it was easier this way.
Be very careful about assuming anything on the Web is solid legal advice (including information here). I know a young American lady who assured me that her research demonstrated that she was allowed to move the UK because she had a British grandparent. Though the UK Home Office Web site makes it clear that this route is only open to Commonwealth citizens, people are allowing rumor and anecdote to lead them to some very strange conclusions.
If you really want to know how to become French, France publishes their laws online. Unsurprisingly, that site is in French, so you may have difficulty navigating it.