(    # this is [0], a hash ref
        [...]
        ),
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No, it isn't a hash ref, it's a list. You should use braces { ... } not parentheses to create a hashref:

my $data = extract();
print $data->{content}->[0]->{A}->[1];

sub extract {
    my %data = (
        site_name => "site name",

        content => [    # array ref
            {    # this is [0], a hash ref
                handicap => [ 0, "asia", "BigSmall" ],
                A => [ "name", "asia_score", "BigSmall_score" ],
                B => [ "name", "asia_score", "BigSmall_score" ],
            },
            {    # this is [1]
                handicap => [ 0, "asia", "BigSmall" ],
                A => [ "name", "asia_score", "BigSmall_score" ],
                B => [ "name", "asia_score", "BigSmall_score" ],
            },
        ],
    );
    return \%data;
}
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Output:

asia_score
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