Four-time champion Vettel said he felt "bitter" to be unable to run in front of his home fans at Hockenheim where he will start Sunday's race from the back of the grid.
Leclerc, who starts from 10th, said it was "a difficult day for the team" and "a shame".
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Toto Wolff of Mercedes, who saw his defending five-time champion Lewis Hamilton take the 87th pole of his career, said: "It's a shame for Ferrari and a shame for Sebastian in his home Grand Prix.
"It's a bit subdued in the crowd and Ferrari have an illness and they need to cure that because we need them for a strong championship."
Ferrari's double failure came as Mercedes celebrated the 125th anniversary of motorsport and their 200th Formula One race entry by sporting 'retro' 1950's uniforms and recovered from a challenging run of practice sessions.
Vettel was hit by a pre-session turbo failure and Leclerc was halted by a fuel system failure that prevented him running in Q3. They will start 20th and 10th on the grid respectively.
"I don't know what happened," said Vettel.
"Something broke with the turbo and that was 'game over' from there. Obviously [it's] very bitter. I think the car is great and lost out on a big chance, but hopefully we'll have a big one coming again tomorrow.
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"I think the car was good... We did some changes also that should have been great for this afternoon, but it wasn't meant to be."
It was the second time in three Grands Prix that Vettel was hit by a technical problem in qualifying.
While Vettel, in desperate need of a strong result after a series of disappointments including a collision with Max Verstappen's Red Bull at Silverstone, was fastest in free practice two, Leclerc had been top in opening and third practice - clear proof that Ferrari were on the pace.
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"I have no idea if it was the same problem as Sebastian for me," said Leclerc. "On my side, it was a fuel system problem. It's a shame, but we'll be trying to understand what happened to try and not reproduce it.
"It's a difficult day for the team, but I hope it'll be a positive day tomorrow. I felt great in the car and the car felt great, all weekend. So it's a shame it ends up like this."
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