First of all, book canon very explicitly disagrees with that statement - in that very same scene in that same conversation, in a statement by the same character (Sirius):
"So. . . so Voldemort could have found out about the tournament?" said Harry. "Is
that what you mean? You think Karkaroff might be here on his orders?"
"I don't know," said Sirius slowly, "I just don't know...Karkaroff doesn't strike me
as the type who'd go back to Voldemort unless he knew Voldemort was powerful
enough to protect him. But whoever put your name in that goblet did it for a
reason, and I can't help thinking the tournament would be a very good way to
attack you and make it hook like an accident."
The bolded part clearly shows that Sirius said that Karkaroff would have to "go back" to Voldemort, implying he wasn't a Death Eater anymore. And moreover, even more important, that he "wouldn't go back" unless a certain condition happened - meaning that he has a choice not to go back.
Second of all, "Death Eater" is a fairly vague term, with a couple of possible exact meanings.
One of the meanings that the the film quote could have used was "people who follow Voldemort".
"Death Eaters?" said Harry. "What are Death Eaters?"
"It's what You-Know-Who's supporters called themselves," said Bill. "I think we
saw what's left of them tonight - the ones who managed to keep themselves out of
Azkaban, anyway."
(src: GoF, Chapter 8).
Under that meaning, clearly: Karkaroff was no longer a Death Eater since he ratted out other DEs to MoM (meaning he wasn't following V. in deed) as well as swore that he wasn't a DE anymore publicly (... in word). He wasn't following in any sense.
Third, another possible way I interpret what Sirius's line meant in the film was more of "once you're evil enough to join DEs, you aren't going to stop being that evil no matter what you say about yourself" - like Malfoy.
If that's the case, again, that interpretation of the meaning is contradicted by book canon, since - unknown to Sirius - both Sirius's brother RAB, and Severus Snape, both stopped being evil internally and meaningfully "stopped" being Death Eaters.
Another possible interpretation is that Voldemort has some way of making former Death Eaters obey his will, even if they wish not to. "You can check out, but you can never leave". Possibly but not necessarily via the Dark Mark, or some sort of unbreakable curse equivalent.
This is not addressed by the book canon, since anyone who "Stopped" being Death Eater was either accepted back in with Voldemort convinced that they never stopped (Snape, Malfoy, McNair); or was killed for the "stopping" immediately (Karkaroff, RAB).
But this lack of examples makes me lean towards the fact that there was no way to "bring a stopped Death Eater back to the fold".
