The problem is you have data on two lines but you are reading the file in a line at a time. You need to remember your state if you want to do it linewise. There are other ways to do it, possibly with split or by setting $/. Here is an example:

while(<DATA>){
    chomp;
    if ( m/^ERROR/ ) {
        $error = 1;
        push @errors, $_;
    }
    else {
        next unless $error;
        $error = 0;
        if ( m/ORA\-00001/ ) {
            pop @errors;  # ignore values and pop off @errors stack;
        }
        else {
            $errors[-1] .= " $_"; # add error string to line msg
        }
    }
}

print "$_\n" for @errors;

__DATA__
ERROR at line 2:
ORA-00002: blah2
junk line
junk line
junk line
ERROR at line 1:
ORA-00001: unique constraint (TBUSOP_1UQ) violated
junk line
junk line
junk line
ERROR at line 3:
ORA-00003: blah3
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