Here are the details: it's a 55 gallon tank, just set up. The tank was running for 24 hours; I dechlorinated the water and put Stress Zyme in the tank. When I got home with my first 4 fish, the filter was dead, and the store is closed. I have somewhere between 4 and 6 inches (definitely less than 6) of fish in the tank: one Angelicus cat, and 3 very small iridescent sharks. With 55 gallons and less than 6 inches of fish I figure I'm safe for a little bit, I'm just wondering how long?
Any information would be greatly appreciated!
Fish can go forever without a filter as long as you ensure there is no ammonia in the tank.
However, even if your filter was working it is not yet in a position to keep the fish healthy. Filters do not come ready to care for your water - you need to make them ready. A filter right out of the box is pretty much useless and you need to do som work before it is useful for your fish.
In short, this is what happens in a tank:
Fish produce ammonia. Bacteria that are living in the water respond to this by attatching to the filter media (sponges, etc) and breeding. As they live and breed, they consume the ammonia and produce nitrite. A second set of bacteria responds to the nitrite and also come to live and breed in the filter. These bacteria turn the nitrite into nitrate. This is called the cycle.
Ammonia and nitrite are highly toxic to fish. Because your filter does not come with bacteria already living in it, you need to provide the correct conditions for the bacteria to grow. You have chosen to use fish for this, which is the traditional way to do it. However, due to the damage caused to the fish by ammonia and nitrite, you need to test your water daily and do very large water changes as needed to ensure the ammonia never gets to a detectable level in the tank. For this reason, some consider the method you have chosen to use inhumane for the fish and would opt for a fishless cycle instead. A fishless cycle is when you manually add ammonia to the tank and don't rely on the fish to do it for you.
This cycle can take up to two months so you can see that it almost doesn't matter that your filter has died as it wouldn't be ready to handle the fish for several weeks anyway. At this stage, you need to find a new filter (or fix the old one) and proceed with daily water testing and water changes.
A fish-in cycle (which you will be doing once your filter is sorted) is a lot of hard work and very dangerous for the fish if not conducted properly.
For now, start doing daily 50% water changes. Buy yourself a liquid test kit such as the API Master test kit that tests for ammonia, nitrite, nitrate and pH and start doing daily ammonia and nitrite water tests. As soon as you see any ammonia (even 0.25 ppm) change a substantial amount of your aquarium water. Even when you get a new filter, keep up with this as it could take two months before your filter can cope on its own.
Alternatively, rehome the fish and perform a fishless cycle which is a lot easier on you and the fish.
Information here on how to set up and maintain a new tank and how to conduct a cycle.
Apologies if you know all of this - just making sure.
Out of interest, why did you choose to buy iridescent sharks? These catfish will grow to around 3 foot long and need to be housed in exceptionally large tanks that are several metres/yards long and contain many thousands of litres. If you want to house three, you need an even larger tank. They are the perfect candidate for a tropical pond but a really, really bad idea in a 55 gallon aquarium.
Unless you can make a cast-iron promise to build a massive, room sized aquarium or heated indoor pond, look to rehome these guys NOW. Try to find a local public aquarium or monster-fish keeper that will take them. If you take them back to the shop they'll just end up sold to someone else with a really small tank. The longer you leave them in your tank, the more likely they are to get stressed as they grow and the harder you will find it to rehome them.
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