It seems fair enough to me, except for some errors. Check the comments for what was wrong:

#!/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__
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