There are a few factors in determining tank size. One of the biggest factors is surface area. The way it works is that fish pull oxygen from the water. The less surface area there is, the longer it takes for the oxygen to exchange from the surrounding air into the water. If you have too many fish breathing, then they use up all the oxygen. As an experiment, put your self completely under a heavy blanket. After a while, you'll notice that it becomes harder to breath. This is because you're burning up the oxygen faster than it can come through the cloth. That's what will happen to your fish. They'll slowly asphyxiate if there isn't a good surface area to fish ratio. You can find many calculators to help you with that. Here is one: Fish Calculator
That's just the oxygen. The next is water quality. Both can be improved by circulation through a filter. Fish are messy animals. They waste large portions of their food, they pee and poo everywhere, and generally make a mess. Just like a person living in their own filth, it will eventually effect their health. They can't help but come into contact with their own waste, so they aren't going to sicken and die like a person might when exposed to cess. However, it's not as hard as it sounds. In the wild, there are all kinds of things helping to process that waste out of the water, from sun, to other animals, to plants (who not only clean waste, but produce oxygen). To fix this in a closed system, you have filter systems. These suck up the solid waste to a degree, filter the water to remove ammonia and other harmful chemicals, and return the water, which agitates it and helps put oxygen back into it. You can add bubblers and real plants to help deal with these issues as well.
The fish will also continue to grow throughout it's life, since fish have something called indeterminate growth. You can look up the adult size of your fish and determine if there is enough room in your tank. I found two good articles here: http://injaf.org/articles-guides/do-fish-grow-to-the-size-of-their-tank/ and http://injaf.org/articles-guides/understanding-fish-stocking-guides/
So to sum it up, there a various factors that determine the fish that can go in a tank. Factors to keep in mind are adult size, oxygen levels, waste removal, and aggression, though that probably doesn't apply to you. I think that at minimum you'll want a tank with some type of filtration system and not one that's stagnant. You could also get away with a smaller tank if you traded in your fish when it started getting too big for the tank. I can tell you that I've done that many times. A pet store will often give you smaller versions since they can make more money off of selling your larger fish. If not, you have a way to get rid of your larger fish and keep the smaller.
Something I'd recommend is getting a 5-10 gal tank (many retail stores and pet stores sell a 5gal hex with a built in filter, and buy some naturally smaller fish. I like the small colorful ones anyway. Guppys are great. They're super colorful and are small. They also breed easily, so you can have babies. That's a different can of worms, though. You can also get neon tetras, platys, swordtails, mollies, etc... You can also get a cool pet like a crayfish or fresh water crab. They don't need much space, just a place to hide (they'll create one if you put an object like a hide in their tank), flowing water ( a filter return is fine), and food to eat. Mine ate fish food, fish waste, and the random muscle I would catch at the river. I'd catch them at the creek and keep them for a couple of years. They molt and get bigger every couple of months. Another cool feature is that they turn colors to match their surroundings. Not like a chameleon, but I caught one that was a 1/4" long and it was jet black. I had white gravel in the tank. He got lighter at every molt and by the time he was 2-3" long, he was blue-white. He started killing my fish and had to go, but he was cool and I caught a smaller one to replace him. They clean fish waste and can be kept with fish if there is plenty of room. They are predators and will hunt them eventually. Good luck.