Global variables are generally to be avoided. If you must use them and have strict vars enabled, you must specify their package as well as their name: $main::foo.

You should take a look at perlmod, perlmodstyle, and perlmodlib. They have some good information about how to write and use modules.

A few things in your code sample jump out at me. I took the liberty of rewriting your code below. I haven't tested it, so there may be errors.

# Save this file as 'MyModule.pm'
package MyModule;  # declare your module to be in a package, 
use strict;
use warnings;      # warnings can help catch logic errors, and typos
use Carp;          # issue warnings from calling code.

my %HANDLE_MODES = (   # replace globals with package scoped lexical t
+hat is NOT accessed outside the file.
   READMODE   => '<',
   WRITEMODE  => '>',
   APPENDMODE => '>>',
);

sub openFile  {
    my $file = shift;  # shift automatically works with @_
    my $mode = shift;  # use strings to match keys in %HANDLE_MODES in
+stead of global variables.

    # check to see if valid mode in call.
    if( exists $HANDLE_MODES{$mode} ) {
        
        # open a lexical filehandle, capturing any errors
        my $fh;
        my $error = open( $fh, $mode, $file) ? undef : $!;
 
        # return filehandle and error string.
        return ( $fh, $error );
    }
    else {
        # if mode is bad issue error message pointing to calling code.
        croak "Illegal file handle mode '$mode'"
    }
}

1;
[download]

Calling code:

use strict;
use warnings;

use MyModule;  # load your module.

# call openFile.
my ($fh, $error) = MyModule::openFile('my/file/here.txt', 'READMODE' )
+;

# Check results 
if ( $fh ) {
    # file opended ok, process file.
    # reading from a lexical handle works just like a global handle.
    while ( defined ( $foo = <$fh> ) ) {
        # do stuff
    }
}
else {
    # File did not open, show error.
    warn "Error opening file- $error\n";
}
[download]

I hope this helps.

Update: I forgot to make sure that the module example returned a true value. Fixed it. I told you I didn't test the code :)