Though locally "canned" food refers to food in a metal can, I understand that in the USA the term "canning" is also used for preserving food in glass jars. I suspect you're talking about metal cans anyway.
If the can is made of metal, the metal can start to corrode (ie react with oxygen) soon after the can is opened, especially since salt in the food can act as a catalyst. For this reason, when I have left over food from a metal can, I always move it into a different (usually glass) container to avoid any contamination, and store it in the fridge. It will last a few days in the fridge. Cover it to avoid accidental contamination and drying out.
Canned foods are (always?) cooked, and in my understanding cooked foods tend to spoil reasonably quickly because they are easily digestible to both us and microbes. I would discard uneaten food after about 3 days, unless it was high in sugar or was acidic.
There's also no reason why you can't freeze the left over food; again, I would move it into a different container to prevent corrosion. Personally, I do this with tomato paste and crushed pineapple (I make ice cubes of tomato paste for use on pizza, and of pineapple for use in toasted sandwiches), surplus pasta sauce, chick peas. I tend to use spare glass jars for things that need defrosting in the microwave, as defrosting in plastic is likely to lead to overheated plastic (and the health risks that come with that).