Liverpool boss Brendan Rodgers set to meet owners with future in doubt
On top of that, a section of the away crowd booed Raheem Sterling even though he was only a substitute.
Brendan Rodgers accepted his future at the club is in doubt after the game.
"I've always said if the owners want me to go, then I go," said Rodgers, who has won two of his last nine games.
"For everybody connected with Liverpool, we are all embarrassed by that result."
It was the first time in 52 years the Reds had let in six goals in the league but Rodgers, 42, insisted he still has "a lot to offer the club", despite winning no silverware during his three years in charge.
Liverpool must also now go through two qualifying rounds if they are to compete in the group stage of next season's Europa League.
"A lot has happened this year which has made the job difficult," added Rodgers, who guided the Reds to second place last season.
"Last season, when things were working well, we had the support of everyone but performances like today do not help that and I fully understand that.
"The fans deserve an apology. They were angry at half-time - and rightly so - because that was awful, absolutely awful.
"They have every right to be angry and frustrated and of course I take full responsibility for that as the manager."
Rodgers dropped Sterling, 20, to the bench at the Britannia Stadium and Liverpool found themselves trailing 5-0 at the interval after a terrible display.
All five goals came in the space of 23 minutes from Mame Biram Diouf, twice, Jonathan Walters, former Liverpool midfielder Charlie Adam and Steven Nzonzi.
Steven Gerrard marked his farewell with a consolation goal but there was still time for ex-Liverpool striker Peter Crouch to head a sixth for Stoke five minutes after coming on as a substitute.
Asked about his rebuilding plans, Rodgers added: "There's an awful lot of work to do and the job is to now go and fix that, and make sure we come back with a motivation greater than ever to push on next season.
"We've finished the season in sixth, which is where we deserve to finish.
"We have to now start, today, on the climb to get back up again."
Rodgers is set to meet the owners to discuss those plans but he has seen his odds to be shown the door shorten.