This is something that has always been on my mind. When I was maybe in first grade, we had a big tank with a lot of goldfish. I wasn't really in charge of their care, but I fed them.
There was one, the biggest, that I named Alice. S/he (because I didn't know, I just assumed they were all female because I was a girl) and a few others would let me pet them when they came up to eat. She'd even let me pick her up out of the water and hold her for a few seconds. I thought this meant she liked me and trusted me, but I dunno.
At some point, some unknown disease killed them all. I always wondered if it because I'd picked her up and passed my human germs on to her. Is it bad to try and play with a fish like that? I recently saw this video gif of someone with their Red Devil, and it reminded me of what I did with Alice: http://giant.gfycat.com/GoodPleasedItaliangreyhound.gif
The fish looks like it's having an okay time, but is this dangerous for them in any way, shape, or form?
Very few diseases can be transmitted between humans and fish.
It is more likely the fish tank did not have proper setup for "a lot of goldfish". Goldfish are messy eaters and they produce tons of ammonia waste. It is recommended to have minimal 20 gallon of water for a single fancy goldfish, and extra 10 gallon for every additional. If it is common goldfish, then the minimal requirement goes up to 40 gallon, and additional 15 gallon is needed for every extra common goldfish. The filter system required for goldfish tanks also must be sufficient enough to handle the ammonia produced by the fish. Ammonia produced by fish through their gills on a constant basis and it is also produced from their organic waste as well as from the leftover fish food in the tank. It is toxic to the fish.
Overfeeding is another common cause for fish dying, because it can produce ammonia spike even in a well established aquarium. With all the kids keep feeding them, it is likely to be one of the causes.
It could be either overstock, under-filtered, or overfeeding, or all of them combined that killed the fish.
As for playing with the fish, most fish will not interact with the human, but many fish which have get used to the feeding routine will display certain behaviors such as following you as you walk by the tank.
the cause of why all your fish died is unknown. trying to remember something that happened years ago is no point. I woudnt worry about it too much. but if all the fish died it was probably do to low water quality.
Some people do handle their fish for various reasons... for example, I've seen aquarists who remove their fish by hand because they feel that netting is dangerous for the creature. Others have done it in order to gently remove a sick fish for treatment. It doesn't typically cause disease to get your human germs on the fish. However, it can damage the fish's scales if you aren't careful, which can allow harmful bacteria into the wound. Others have already answered your question about your goldfish dying, so I won't touch that, but yes, goldies CAN be handled... but only with extreme care.