Hannah Barnstable is determined to achieve what shouldn’t even be possible — bootstrap a new breakfast cereal company.

She has jumped into a nearly $10 billion-a-year business in the United States that is dominated by just a handful of players, the likes of Kellogg Co., General Mills and Post. If any of them cared enough to try, they could easily crush Barnstable’s Seven Sundays LLC.

But three-year-old Seven Sundays just started selling in Target. It could be that Barnstable is heading toward the kind of entrepreneurial success that always seems to be impossible — an undercapitalized upstart grabbing a profitable share of a market from global giants.

Yet it’s the kind of thing that happens all the time. While the details differ, the basic story usually is the same. The big companies are a little too wedded to the way things have always been done.

“I know there’s a lot of big companies in the breakfast category, but everything was just getting cheapened and cheapened,” Barnstable said. “And breakfast was not getting the credit it deserves as the most important meal of the day. I just had this passion, that somebody needed to come out and reinvent the category.”

Her idea is to get people to treat every day just like a Sunday morning, with a healthy breakfast if not a leisurely one. Her first product was muesli, a mix of rolled oats and other grains along with nuts, seeds and dried fruit.

There is nothing particularly innovative about muesli, but what makes the product sell is what consumers have to compare it with in the cereal aisle.

“Everything is very engineered,” she said of the big companies’ product lines. “So cereal isn’t even food anymore. Instead of trying to engineer food, I am almost reverse engineering it, to go back to the basics.”

What’s interesting is that her insight into the U.S. cereal market is shared by deeply seasoned securities analysts. The cereal market has been in decline, and “engineered” products are part of the reason.

In a research report last week, it took Sanford C. Bernstein analyst Alexia Howard just one long paragraph to completely take apart Kellogg for mismanagement of its U.S. cereal business.

“There does not appear to be any quick fix,” she wrote, adding that Kellogg’s new proposed fix — smarter marketing — isn’t likely to work any better than last year’s proposed fix did. “The company needs to overcome increasing consumer concerns about gluten, [genetically modified organisms], artificial/complex ingredients and sugar in order to win consumers back to the category, which may prove more challenging than just a new marketing message.”

Her research shows that processed foods are falling out of favor, and her recent survey showed that 16 percent of consumers bought more boxed cereal over the last five years — but 27 percent bought less.

The big companies have responded with more than fresh advertising, of course, but it’s not a stretch to say it’s been more than 100 years since the period of greatest innovation in the breakfast cereal business.

Here’s a recent story of innovation, big-company style: Since 2012, Post has been on a roll, starting with two new Honey Bunches of Oats Fruit Blends, then new Honey Bunches of Oats Greek Honey Crunch and Honey Bunches of Oats Greek Mixed Berry. And last year there were three versions of Honey Bunches of Oats Granola.

Post just keeps playing variations on this theme. Its most recent launches were for Honey Bunches of Oats Morning Energy in two flavors, including chocolaty almond crunch.

If Post wants to keep customers from switching to what they perceive as healthier breakfast options, launching a chocolaty almond crunch version of any “energy” cereal doesn’t seem to be going in the right strategic direction. It came from a lab, not a test kitchen. And what, exactly, is “chocolaty”?

The companies have tried to market products closer to what Barnstable is selling, too. More than a decade ago, Kellogg acquired the natural-cereal player Kashi Co.

Kellogg’s CEO late last year said that the Kashi natural foods brand was no longer perceived as being much different from a more ordinary consumer brand. That’s one explanation for why Kashi sales are falling.

Another is that Kellogg damaged its brand by mishandling a controversy over GMO ingredients. Meanwhile, the company is likely feeling the effects of dragging all sorts of products under the Kashi label.

Barnstable said a key to the Seven Sundays brand is authenticity. She observed that it couldn’t have helped Kashi’s brand for longtime customers to see Kashi frozen chicken Florentine dinners for sale.

She isn’t planning anything of the sort, of course. But even with an opportunity to carve out some sales from conventional competitors, “this has been 10 times harder and 10 times more expensive than I thought it would be.”

A spot on a shelf, with the label facing the consumer, is expensive, scarce real estate. So-called slotting fees are common. As a start-up, Barnstable says, she more often gets demands for what are known as “free fills.” That means giving away the first cases for the retailer to stock the product.

The challenge of getting into stores is why longtime food industry investment banker John Trucano summed up what Barnstable needed with just one word: “Perseverance.”

She sold her first bag at a farmers market in June 2011, got products into Lunds and Byerly’s in 2012 and last year launched in Whole Foods, Cub, Hy-Vee and other retailers.

Last year, Target approached her to discuss its plans for products marketed by natural food companies like hers. “That to me was really telling, somebody like Target, because they are really tied in with the big cereal companies,” Barnstable said. “They are almost partners with them.”

Seven Sundays products began appearing on the shelves of Target stores in Minnesota in mid-January.

Barnstable is 32, and she said she’s still just getting started. But she no longer thinks selling Seven Sundays to one of these big companies will be in her future. She now envisions a longer run with lots of products in addition to cereal.

“Our goal is to reinvent breakfast in the U.S.,” she said. “You think of all the things in the breakfast aisle and kind of how bad they are, like the Pop-Tarts and that stuff. And we think, ‘How can we improve this?’ ”

Sounds like an opportunity.