You could wrap that pattern in a subroutine if the error code is always the same:

sub perform {
    my( $code )= @_;
    eval { $code->() };
    if( $@ ) {
        ...
        # error handling
    };
};

while() {
    perform(sub{
        # connect-to-database
    });
    perform(sub{
        # munge-data
    });
    perform(sub{
        # write-report
    });
}
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Also see Try::Tiny.

The only drawback of using eval {} over not using it is that each eval { ... } is a tiny bit slower than the code without it. If you are doing anything else in your code, that slowdown is usually not worth optimizing away.