If you want to walk the list in a random order ("without replacement" in the vernacular), then you need something to maintain state, and be aware of the size of your list (both initially and at each call).

Your claims of no spare memory make this difficult. Either you need a scheme that implicitly keeps track of which elements have already been returned, or you need to explicitly keep track of them.

On the explicit front, you can keep a record in a bit vector, with one bit for each element already used. However, to get the last element of a large list, using a PRNG will take a long time.

On the implicit front, you need an iterator thats reasonably random, yet is exactly uniformly distributed regardless of the size of your list. This makes me think of a prime modulo operation:

#!/your/perl/here
use strict;
use warnings;

{ # closure for "static" variables
    my $mult   = 2**15-1;
    my $offset = 76543;
    my $state  = 0;
    
    sub get_rand
    {
        my $list = shift; # ref
        $state = ($state*$mult+$offset)%scalar(@{$list});
        return $list->[$state];
    }
    
    sub init_get_rand
    {
        my $list = shift;
        $state = rand(scalar(@{$list}));
    }
}

my $list = [0..17];

for (1..2)
{
    init_get_rand($list);
    for ( 1..@{$list} )
    {
        print get_rand($list), " ";
    }
    print "\n";
}

__OUTPUT__
8 9 16 11 12 1 14 15 4 17 0 7 2 3 10 5 6 13
2 3 10 5 6 13 8 9 16 11 12 1 14 15 4 17 0 7
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Now, I'll be the first to admit I haven't checked on good values for $mult, $offset, or whether lists of certain sizes will cause you to miss values, etc. But you should be able to look up a good PRNG on your own ;)