No trouble at all. Yes, I do. I am technically unpaid staff (a volunteer), but I essentially get to go to the National Aquarium twice a month for free. Plus my staff ID gives me 24/7 access as long as I go through the staff entrance.
I can't speak for other aquariums, but I know we have some pretty strict rules regarding who is allowed in nonpublic and backup areas. To show a non-staff member a backup or equipment area, we need written consent from our supervisor (our supervisors must also get clearance from THEIR supervisors). Unfortunately, we also are not allowed to share pictures or videos of the backup or equipment areas without written consent from supervisors. I'm not sure if this is just the National Aquarium or all aquariums. During training, they impressed upon us that with the National Aquarium being such prominent institution, we have special publicity and liability concerns, which is part of the reason for the restrictions.
The National Aquarium offers guided tours for an additional $15 (on top of admission) which will show you some of the backup and prep areas where we care for animals. You don't see everything though.
I might ask my supervisor one of these days if I can take a few pictures of our equipment area for the Living Seashore to share. The equipment is pretty impressive. We have a community tank which I mentioned above of about 1,800 gallons, a jellyfish touch pool of about 900 gallons, a main touch pool of about 3,600 gallons with rays, skates, horseshoe crabs and whelks, and a few smaller backup systems. The setups are actually really interesting. All the large displays have sand filters, UV sterilizers, ozone, bioballs, and a huge 8" wide, 6 foot clear acrylic skimmer. I'd be happy to answer any questions, I just can't share pictures sadly.