Former eBay CEO Meg Whitman today announced plans to run for governor of California, to replace Arnold Schwarzenegger in what’s sure to be a hotly contested 2010 race.

And though it’s rare for Fortune 500 bosses to become the CEO of a state, California’s disastrous financial condition gives Whitman, a political rookie, a clear opportunity to sell herself on management skill and discipline.

If ever California needed that, it is certainly now. The employment picture is terrible: California’s 9.3% unemployment rate is among the highest in the U.S. — and on top of that, the state, to preserve shrinking finances, has begun furloughing its own workers every other Friday. Meanwhile, the budget outlook is dire: Standard & Poor’s last week lowered California’s credit rating to the lowest among all 50 states. Facing a deficit expected to reach $42 billion by mid-2010, Governor Schwarzenegger has been clashing with both Democrats and his fellow Republicans about how to save California from default.

Whitman, 52, a Republican, hadn’t intended to dive into politics after ending her decade-long run at eBay last March. But after getting pulled into last year’s Presidential race — first as finance chair for Mitt Romney, who had once been her boss at Bain & Co., and then as co-chair of John McCain’s failed campaign — she picked up the political bug. In a speech last September to the Commonwealth Club of California, she said she was struck by how much a campaign, intense and fast-paced, “is just like a Silicon Valley start-up.”

Meanwhile, she was fretting about the direction of the state that she moved to in 1998, when eBay (2008 revenues: $8.5 billion) had 30 employees and $4.7 million in revenues. “The question I’m asking myself more and more these days is, if eBay were getting started today, would Silicon Valley and California still be the best place for it?,” she asked in the Commonwealth Club talk. “Is California still the best place for innovation and job creation in America?”

Her implied answer, of course, is No — and Whitman (who, with today’s announcement, is holding off on press interviews) is likely to run on a campaign that is critical of the way Schwarzenegger, who is prevented by term limits from running for a third gubernatorial term,  and the legislature have managed the state. The former Terminator has disappointed fellow Republicans who expected him to rein in spending. Whitman, speaking to a group of district attorneys last month, said, “If California were a business, we’d be out of business.”

No question, Whitman — who recently gave up her board seats at Procter & Gamble , DreamWorks Animation and eBay — is entering a contest that will test her toughness. First she needs to get past her key GOP rivals, former Congressman Tom Campbell and California insurance commissioner Steve Poizner, a former Silicon Valley entrepreneur who sold his company, Snaptrak, for a reported $1 billion to Qualcomm nine years ago. If she cops the Republican nomination, Whitman could face Dianne Feinstein, the popular California senator who has hinted about running ever since she lost a tight governor’s race to Pete Wilson in 1990. Now 75 years old and ever formidable, Feinstein has lofty credentials (she chairs the Senate Intelligence Committee) and a good relationship with California’s business community.

If Feinstein decides that staying in the Senate is safer career-wise (which it surely would be), the Democrats will likely nominate another big name. Among the possibles: California attorney general Jerry Brown, who served as governor from 1975 to 1983, San Francisco mayor Gavin Newsom, or Los Angeles mayor Antonio Villaraigosa, if he chooses to run for governor.

As for another well-known Californian who preceded Whitman as Fortune‘s No. 1 Most Powerful Woman in Business, Carly Fiorina, the former CEO of Hewlett-Packard , is said to be considering a run for Barbara Boxer’s U.S. Senate seat in 2010.