Two judges in London rejected a judicial review application.
Bamber challenged a refusal by the Criminal Cases Review Commission (CCRC) to refer his case back to the Court of Appeal as a miscarriage of justice.
The CCRC, an independent body which investigates possible miscarriages, rejected Bamber's last appeal in April.
The decision on Thursday follows a single judge rejecting Bamber's application for permission to seek judicial review of the CCRC's decision after he studied the case papers in private.
Bamber made a renewed application dealt with by Sir John Thomas, president of the Queen's Bench Division, and Mr Justice Globe.
Announcing the decision, Sir John said that having looked at the approach taken by the CCRC in the case he could not see "any way" in which a challenge could be made to the decision reached.
"It seems to me that a challenge is impossible to mount," he said.
The 51-year-old who is serving a whole-life term for the 1985 killings at a remote Essex farmhouse, has always protested his innocence.
In April, the CCRC said that despite a lengthy and complex investigation, it had not "identified any evidence or legal argument that it considers capable of raising a real possibility that the Court of Appeal would quash the convictions".
Bamber and two other killers have also started an appeal in the European Court of Human Rights against spending the rest of their lives in prison, claiming a breach of human rights.
In a statement posted on his website, Bamber said: "It appears that the threshold for my case to be referred to the Court of Appeal is much higher than in most cases but that doesn't make me any less innocent.
"The law, it seems, simply does not apply if it assists me in proving that I am wrongly convicted."
Killer Jeremy Bamber has failed in his latest High Court action to overturn a conviction for murdering five relatives 27 years ago in Essex.