How about reading a block at a time and spliting that?:

sub genBufferedGetNum {
    my @buf = do{ local $/ = \4096;  split ' ', scalar <>; }
    my $leftover = pop @buf;
    return sub {
        unless( @buf ) {
            unless( eof  ) {
                @buf = do{ local $/ = \4096; split ' ', $leftover . <>
+ };
                $leftover = pop @buf;
            }
            else {
                die 'premature eof' if $leftover != 0;
                return $leftover; # last number
            }
        }
        return shift @buf;
    };    
}

my $getNum = genBufferedGetNum();

while( my $num = getNum->() ) {
    ## do stuff
}
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