Me thinks you have the wrong end of the stick. You seem to want to match 7000 or 7001 or 7002 or 7003, but you have put that in a character class so what is being matched is one of '0', '1', '2', '3', '7', or '|'.

You also have a problem with your '(' and ')' - they are capture parenthesis which is not what you want. Probably what you really want is:

use warnings;
use strict;

my $full_text = "(7001) - This is some text";
my $check_str = '\((?:7000|7001|7002|7003)\) - This';

print "full_text has the check_str\n" if $full_text =~ /$check_str/;
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Note the use of ' rather than " so you don't have to quote the \ (as in \\) to quote the '(' and ')' and that a non-capture group '(?:...)' is used for the alternate group. Because the 700 is common $check_str could become '\(700[0123]\) - This' - using a character class to match the last digit.

Take a look at perlretut, especially at the 'Matching this or that' section and the 'Using character classes' section.

Update: fix various typos and omissions