Last Friday, the Disney Store in Times Square was the saddest place on Earth.

Kids streamed in, one after the other, talking excitedly to their parents about Elsa and Anna, the stars of Disney’s hit “Frozen,” the top-grossing animated film of all time, with more than $1 billion in worldwide box-office receipts.

But their smiles quickly melted.

“We’re all sold out of ‘Frozen,’ ” a Disney sales associate said for the 200th time that day. “Except for this,” she added, pointing — weakly — to a paltry stand decorated with five pairs of Anna boots, a handful of floral frocks emblazoned with a “Frozen” logo and four “Frozen”-themed ballet flats large enough to house Dennis Rodman’s feet.

“I can’t believe in a great big store, this is all they have,” lamented Pauline McDougal, who was visiting from Scotland. Her daughters, 11-year-old Lauren and 8-year-old Megan, had their hearts set, respectively, on the Elsa dress and Elsa doll.

“You’d expect more in New York,” McDougal added.

It’s official. “Frozen” fever has swept the world. The only problem is, the merchandise is sold out everywhere.

Since the movie’s release on DVD in March, “Frozen” merchandise has been selling so fast, Disney had to institute a two-item limit on all goods last Wednesday (not that there’s anything to buy).

Maternity-wear designer Rosie Pope says her pal — who works at Disney, no less — recently shelled out $1,200 for an Elsa doll on eBay after she promised her daughter one for her “Frozen”-themed birthday party.

“By this point, she didn’t care about the price,” says Pope. “She didn’t want to disappoint her daughter.”

It’s a buy-or-die mentality. And one that Upper West Side mom Shannon Russo-Pollack admits to adopting. While at Walt Disney World last month, Russo-Pollack trekked to more than 42 stores in search of the elusive Elsa dress for her 6 ½-year-old daughter, Summer.

“They were totally sold out,” says Russo-Pollack, who owns Dasha Wellness and Spa in NYC.

That’s when Russo-Pollack’s husband, Dr. Darren Pollack, plopped down $480 on Amazon to preorder two dresses (he didn’t know which size to buy), in addition to more than $350 on other “Frozen” paraphernalia.

“I said, ‘What are you, crazy? Keep looking!’ He said, ‘I promised her. How many more stores can we go into?’ ” recalls Russo-Pollack. “It was crazy, but totally worth it. My daughter’s eyes lit up when she saw the dress.” (Russo-Pollack will be donating the extra size 6 dress to the Leukemia and Lymphoma Society.)

Experts liken the frenzy to the 1980s Cabbage Patch Kids craze.

“We’re now at the stage where the demand is almost being driven by the scarcity because of the social status attached to being able to find it,” says Sean McGowan, a toy industry analyst at Needham & Co. in NYC. “Being someone who had a Cabbage Patch [Kid] meant you were loved more than the others. It was social status and elite achievement that came with finding this rare gem.”

According to Times Square’s Disney Store employees, every other customer who walks into the flagship is on the hunt for “Frozen” gems.

Needless to say, “sold out” doesn’t always sit too well.

“People have gotten into physical fights in the morning,” says one Disney Store employee, who asked not to be named.

“The kids cry, but the parents are the problem. They try to guilt us, say their daughters are sick. They have no shame. But I can’t make it magically appear!”

Donna Ladd, who writes a blog called Motherburg, didn’t even realize what a hot commodity “Frozen” had become until after she snagged an Olaf doll during a recent business trip to Italy: “I was with another mother, and we passed the Disney Store in Venice and we saw the ‘Frozen’ shop and she went crazy.”

Ladd brought home the stuffed snowman for her 4 ½-year-old son, Charlie.

Chaos ensued.

“Anywhere I was, at the Met, at the supermarket, all the mothers were going crazy screaming, ‘Oh my God, I can’t believe you got it!’ ” says 43-year-old Ladd, who lives in Greenpoint, Brooklyn. “They were asking me if they could borrow the doll for a few days . . . I feel like I had a bag no one else could get.”

Things got so intense that Ladd became wary of letting Olaf out of the house: “It’s causing so much angst for the kids and parents, and also kind of making them sad,” she says.

Disney message boards are rife with angry mothers lambasting the company for being ill-prepared; some are even accusing the company of orchestrating the shortage to ramp up interest. But McGowan says the latter is nonsense: “There’s no chance Disney doesn’t want to be selling more.” Retailers just made the mistake of playing it too safe after being burned by an excess of inventory from lackluster movies including “Tangled,” “Brave” and “The Princess and the Frog,” according to McGowan.

“We knew this movie was a winner,” says Erin Barrier, a Disney Store spokesperson, “but it overperformed so significantly that now we’re doing what we can to get in more product as soon as we can.” (Barrier says the Elsa dress should be back in “limited quantities” by early May and “back fully in stock in July and August.”)

But most moms aren’t willing to wait.

Lyss Stern, founder of Divalysscious Moms, a luxury lifestyle company, is already stocking up for her 6 ½-year-old son’s “Frozen”-themed birthday party — in July — because “I know I’m not going to be able to find anything,” she says.

Leslie Venokur’s friend was in such a bind for her daughter’s “Frozen” birthday bash last Sunday that the pal shelled out $150 on a homemade Elsa costume for her child from the crafts site Etsy. “It didn’t even look like Elsa,” bemoans Venokur, co-founder of the Big City Moms site.

In an attempt to override the price-gougers, mothers have set up Facebook groups, such as “Unfrozen Trading Friends,” in which approved members can exchange tips on where to score “Frozen” merchandise and sell their toys, at cost, to one another.

When one group member posted that Disney canceled her Elsa doll order, D.M., a lawyer who lives in New Canaan, Conn., FedEx’ed the distraught mother a spare Elsa doll she owned.

“She was so upset because her daughter is chronically ill and going through chemo,” says D.M., who asked that only her initials be used for professional reasons.

With a Netflix release of “Frozen” scheduled for Tuesday, even rentals of the movie have been hard to get recently — the “Frozen” wait-list at the Scarsdale, NY, library is 702 deep (with just 5,418 households, according to the 2010 census).

The biggest “Frozen” fanatics admit, though, that it might be time to just let it go.

“You want your kids to be happy, but at the same time, what are you willing to do, what are you willing to pay?” asks 31-year-old Upper East Sider Nicole Ross, who has a 3-year-old daughter, Sydney, and another child on the way. “It’s really dumb. We should probably just learn to say ‘no’ more often. But it’s so difficult when they are so obsessed with the current movie.”

“The sad thing is, when all this stuff gets restocked, they’ll be on to the next thing,” admits Venokur, who still hasn’t managed to get her daughter the Elsa dress.

“Until ‘Frozen 2’ comes out.”