How’s this for an impossible gig? Arrive at what was once one of the most promising companies on the planet, a company that still has 700 million customers in over 20 countries, but one that has systematically ceded ground to competitors as if it were a religious obligation. It is now your task to turn the culture around, placate rabidly activist shareholders, and somehow reinvent the internet in the meantime.

This is what the newly arrived Marissa Mayer must do when she checks into Yahoo this week. She comes to the web’s fourth largest site at an especially troubled time in its especially troubled history: new chief executive Scott Thompson, a former president of PayPal, was unceremoniously canned for inconsistencies on his resumé. 

Here’s how things have progressed at Yahoo. Founder Jerry Yang, one of the web’s bright young things back in the Bill Clinton administration, envisaged a portal where people would come online to read on computer as they once did on paper. Yahoo became enormously popular very quickly, and it brought in lots and lots of advertisers to go with all the eyes. But besides just sitting there and looking mediocre, Yahoo didn’t really do anything. No search engine, no proprietary media tech, no real commitment to R&D. Where Google rolled out aggressive new product lines in order to evolve the business, Yahoo did no such thing. Until it was too late. Then it tried to do everything at once.

By the time Yang was ensconced in the CEO position in 2008, Yahoo was caught in a bullhorn formation between Microsoft and Google. Frustrated by ongoing and increasingly less cordial merger talks, Microsoft made a hostile bid for $47,5-billion, which Yang and company battled for four months. Enraged shareholders, expecting a dividend or two, helped him out the door.

In came Carol A. Bartz, a tough, knuckle-fighting tech exec who founded Autodesk. In her seven seconds at the head of the company, she tried to focus on the company’s “strengths”—its batch of media sites covering news, sports, entertainment and everything else under the digi-sun. She firmed up a deal with ABC, but a botched Microsoft search-engine partnership proved too much for investors, who gave her the ritual boot.

In came Thompson, who again tried to subtly change direction, while doing what Yahoo does best: downsizing. He insisted that he would concentrate on three core groups: consumers, regions and technology, and no one knew what in heck he was talking about. Almost all of the company’s earnings in the last several years came from the chainsaw: Thompson let 2,000 employees go, and started concentrating on “content”—the idea, popular in some Yahoo circles, that producing its own stuff would be a sound idea. But after botched sales of Chinese partner Alibaba and Yahoo Japan, Thompson had punched his own ticket for the little leagues, and is now the chief executive of ShopRunner, a company that provides the same two-day shipping services as giant Amazon.

Yahoo, for all its manifest problems, still brings in revenue. Just not as much as it should. Net income for the second quarter of 2012 was down 4.2%, to $228,5- million. Revenue declined slightly to $1,22-billion, and the company makes about 18 cents a share. These aren’t bad numbers by any stretch, but they still represent a massive disappointment, given the company’s cash in hand, resources and customer base. So, what’s a girl to do?

Mayer has just about the best pedigree of anyone Yahoo has poached into a senior position in a very long time. She is an old hand at Google—insofar as anyone in Silicon Valley is “old”—having trained as an engineer. She became instrumental in designing and guiding the look and feel of the iconic site. All that white space, all the untrammelled, snowy lengths of computer screen that add to ease of use? Thank you, Marissa Mayer. To visit Yahoo for three blinding seconds is to understand why the board was interested in paying her what could amount to $100-million over five years. That’s Lebron James money. 

Already worth in the region of $300-million, Mayer isn’t in it for pay day. She’s signed up for the challenge of transforming what could have been—should have been—one of the world’s reigning iconic brands, à la Coke, or, say, Google, into an internet monolith. The task is all but impossible. Chic, immaculately turned out, with geek credentials that could blow out a roomful of Dark Knight enthusiasts, Mayer does have the mojo. But what exactly is Yahoo? 

Mayer has said that when she started at Google, “everyone mistook the internet for Yahoo.” Which is, in fact, part of the problem. Its product and site rollouts have been so scattershot and all over the place that ubiquity was substituted for a business plan. Does the site create content, deliver it, augment it, or merely filter it through new or acquired social network sites (of which Flickr is by far its most successful)? No one knows the answers to these questions. Yahoo hasn’t lost ground as a brand—its recognition has remained steady over the course of the past decade— but it has all but imploded as a product. 

So, that question again: what’s a girl to do? No clue, mostly because the debate around Yahoo has been put on hold following a Tweet Mayer knocked off last week: she’s pregnant. This has cracked open the particularly fraught debate about working women and maternity leave: Mayer told Fortune that her break would be a few weeks long, and she would work throughout. You get where she’s coming from—there hasn’t been a human infant born on Earth as defenceless and mewling as the company she now helms. The kid will be fine. The company needs a miracle. DM

Read more:

• “Maternity leave? More like a pause” in the New York Times
• “A Yahoo search calls up a chief from Google” in Dealbook

Photo: Marissa Mayer, then vice president of search products and user experience for Google Inc, unveils “Google Instant” during a news conference in San Francisco, California September 8, 2010. REUTERS/Robert Galbraith