The holders suffered a 2-1 defeat to Brighton and Hove Albion at the AMEX stadium despite taking the lead through Harvey Elliott.
Lewis Dunk's fortuitous equaliser and Kaoru Mitoma's stoppage time winner saw Jurgen Klopp's side surrender their grip on the trophy they won last year.
Liverpool were in the hunt for four trophies last season and ended up winning two but are way off a title charge and the Champions League places this term.
21 points separate Liverpool and leaders Arsenal while the Reds are 10 points off the top four.
Liverpool were hoping for a reset after the World Cup but have won just one of their six games in 2023.
"This season has been nowhere near good enough," Robertson told ITV Sport.
"At the start of the year, we wanted a fresh start, but that hasn't happened – we've been worse. In the league, we haven't been good enough and now we're out of both cups."
The left-back added: "It's really disappointing. You can't put your finger on one thing that's gone wrong. It's more than that. It's about trying to put a performance together where all these things click and that's proving difficult right now.
"You can tell we're not as confident in front of goal, and defensively we're still a wee bit open in certain areas.
"We were on the back of two clean sheets, which was positive, but today, two disappointing goals and they had a couple of big chances as well.
"We need to try and get the confidence back. It's easier said than done but that's the only way we can pick up results. You have to be able to score goals and you have to be able to keep clean sheets.
"We're not managing that just now and we need to do it. We keep saying that and it doesn't help the fans. I feel sorry for them, the way we're putting on a show for them just now.
"We let them down again and we're obviously disappointed to be falling out of the cup."
