An appropriate idiom for this kind of thing is to use die with a special token:

use constant END_REQUEST => "END_REQUEST\n";

sub end_request { die END_REQUEST }

REQUEST: while(accept()) {
    eval {
        # process request
        1;
    } or do {
        next REQUEST if $@ eq END_REQUEST;
        die $@;     # propagate real errors
    };
}
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