VILLARREAL, Spain — In the corner of the Estadio de la Céramica, the one left totally at the mercy of the elements, the fans started to unfurl their scarves. On the scoreboard at their backs, the clock had ticked beyond 90 minutes. On the field in front of them, Villarreal was on borrowed time in the Champions League.

It was then that they started to sing. As Liverpool enjoyed a rare moment of calm after a storm of an evening, and put the finishing touches on its 3-2 victory, the rest of the stadium noticed what was happening in the corner, and picked up the tune. They held their scarves aloft, too, a gesture of defiance, and loyalty, and gratitude.

And then, when the whistle blew and it was all over, as Villareal’s players walked mournfully around the stadium, heads bowed and eyes raw, the tempo quickened. The scarves started to twist and to whirl, the mood shifting from regret at what had been snatched away to celebration of all that remained. In the pain, they found pride.

Indeed, quite how much it hurt was, perhaps, the best measure of quite how close Unai Emery’s Villarreal had come. This team was not supposed to be in the Champions League semifinals, not really; the very structure of European soccer’s elite competition is built to make it vanishingly unlikely that a team of its stature could travel this deep into the tournament.

Villarreal was certainly not supposed to stand a chance going into the second leg. It had, by common consensus, been summarily dispatched at Anfield last week, its limitations exposed by the depth of Liverpool’s resources and the scope of its firepower and the sheer gravity of Jürgen Klopp’s team. The return leg was, more than anything, an administrative hurdle to be cleared, a form to be completed.

Villarreal, the town, is a curious place to stage a game of this magnitude: a satellite of nearby Castéllon, more than anything, quiet and refined and, after a day spent under a torrential downpour, almost entirely deserted. Snatches of songs, in both English and Spanish, echoed around the streets.

If the sense of occasion that ordinarily accompanies the most seismic games in Europe’s calendar was missing outside, it was palpable inside. For the first time, Villarreal had arranged a mosaic: a blue submarine against a yellow background, the club’s slogan, Endavant, picked out in giant letters. The public address announcer talked about believing in comebacks.

Any doubters would have been converted within three minutes, as Boulaye Dia tapped home from Étienne Capoue’s not entirely intentional cross, and the Céramica seemed to melt. All of a sudden, everything felt possible. Liverpool, so seamless and so smooth oin a 2-0 victory six days ago, struggled to complete a pass.

By halftime, its rhythm had been broken and its confidence sapped and then, just when it thought it might make it through, its advantage had disappeared completely. Capoue crossed, on purpose this time. Francis Coquelin headed home. Villarreal’s bench emptied onto the field, coaches and substitutes and sundry assistants all scarcely able to believe what they were seeing.

At that moment, tied at 2-2 halfway through the second leg, Villarreal’s players stood within touching distance. The final was there, right there, and they could seize a place within it. Villarreal would be the smallest town, by some distance, to send a team to the biggest game in soccer.

In an era defined and designed by Goliath, it would be this team, constructed on a shoestring, that did what Ajax and Monaco and RB Leipzig could not and made it all the way. And they could do it by etching their own entry in the Champions League’s ever-expanding book of eye-watering comebacks, a miracle to call its own, just like Barcelona (2017), Roma (2018), Liverpool (2019) and Real Madrid (passim).

Hope and belief exist at different points on the same axis. Villarreal, in the space of 45 minutes, had traveled all the way along it.

And then, just when it was there, within their grasp, it was taken away. Klopp took off one $45 million forward, Diogo Jota, and introduced another, Luis Díaz. The switch changed the momentum irrevocably. Trent Alexander-Arnold hit the bar. Díaz tried a spectacular overhead kick. And then Mohamed Salah slipped Fabinho through and his shot squirmed through Géronimo Rulli’s legs. In that moment, it was all over.

Five minutes later, Díaz had scored, drifting in to head a cross under Rulli. Five minutes after that, Sadio Mané had put Liverpool ahead on the night, latching on to a pass from Alexander-Arnold, darting past Rulli as he charged out of his goal and into midfield, and then calmly rolling the ball into the net.

Perhaps, in hindsight, it would have been easier had Villarreal not heard that siren call of possibility. Perhaps it would have been easier to go quietly, to succumb to the inevitable. That might have hurt less. But then the journey is not defined by the destination.

Villarreal beat Juventus in Turin in the round of 16. It silenced Bayern Munich in the quarterfinals. And it produced 45 minutes that saw Liverpool — a team now on its way to a third Champions League final in five years, a team pursuing an unprecedented and scarcely possible clean sweep of trophies — so scrambled that when Klopp asked his assistant, Peter Krawietz, to identify a “single instance” of good play from the first half and show it to the players for inspiration, he came back and told him there was nothing to be found.

And it did it all on a budget that is a fraction of its rivals, in an ecosystem in which the big beasts consume most of the oxygen, and with a team patched together from the discarded and the dismissed. There was a common root to the pride and the pain: At times, an aching wound can feel like a badge of honor.

“Soccer is beautiful,” Villarreal’s captain, Raúl Albiol, said. In time, he knows, what will matter is not that Villarreal fell 45 minutes short of a Champions League final, but that it came to be in a position to fall 45 minutes short of a Champions League final.

“This was a defeat,” he said, “but we’ll always remember this run.”