#!/usr/bin/perl

use strict;
use warnings;

my %hash = (
            # use ampersands, not dollars
            func   => \&func,
            func2  =>  \&func2,
           );

while () {
    # don't need ampersand to call function
    menu();
    print "select: ";
    # use my to declare $string
    chomp(my $string = <STDIN>);
    # use exists to stop from auto-vivifying hash elements
    if (exists $hash{$string}) {
        $hash{$string}->();
    } else {
        print "No such command: $string\n";
        sleep 1;
        system("clear");
    }

}
sub quit { exit; }

sub func { print "running func\n"; }
sub func2 { print "running func2\n"; }


sub menu {
    while( my ($k, $v) = each %hash) {
        print "$k ";
    }
    print "\n";
}

__END__
[download]

Someone already pointed out that you need to define your dispatch table a little differently. Your example is creating a hash of scalar references, instead of a hash of subrefs. I won't belabor that point since it's already been made.

I did want to mention that your dispatch table needn't use named subs at all. Consider the following code:

my %dispatch = ( 
    DoThis => sub { print "You asked me to do this.\n"; },
    DoThat => sub { print "You asked me to do that.\n"; }
);
[download]

For trivial subs, defining them inline like this can be a simple, convenient, and readable solution.

Another solution is the switch construct:

SWITCH: for ( @commands ) {
    m/^This/ && do { print "You asked me to do this.\n";
                     next SWITCH;
                   };
    m/^That/ && do { print "You asked me to do that.\n";
                     next SWITCH;
                   };
    m/^Another/ && Another_Way();
    m/^Quit/ && do { print "Goodbye.\n";
                     last SWITCH;
}

sub Another_Way {
    print "You asked me to do it another way.\n";
}
[download]

Of course there are other ways to do it as well... This is Perl after all.