Real, who had won four of the last five editions of the competition, are almost certain to end the campaign empty handed, having been knocked out of the Copa del Rey in the semi-finals by Barcelona last week and having fallen 12 points behind the Catalans in La Liga.
"I think I have never felt this awful, I do not know how to explain it," Carvajal said after Ajax completed a 5-3 aggregate victory.
"This week everything has gone against us. Tonight we cannot look for excuses. Our opponents were better and they deserved to go through. We have to show our faces and not hide, our season has been a piece of shit."
Real, whose captain Sergio Ramos was suspended after intentionally getting a yellow card in the first leg, conceded two early goals as Hakim Ziyech and David Neres scored for the rampant visitors, who had also dominated the first leg despite losing.
The outstanding Dusan Tadic made it 3-0 with a thumping strike in the second half and although Marco Asensio pulled a goal back for Real, Lasse Schone made it 4-1 with a free kick which flummoxed goalkeeper Thibaut Courtois.
"It's never easy to go into a game after two defeats to Barcelona, today we came here with a clear idea that we wanted to win, then two mistakes meant we were two goals down," Carvajal added.
"The start of the second half was good but the third goal killed us."
(Reporting by Richard Martin; Editing by Toby Davis)
