I am wondering about a kind of padding made from asbestos that is sometimes found in the back of old radios to protect the wood or cardboard on the back from the heat of old vacuum tubes. It sort of resembles under-carpet padding.
Is this padding, and the dust from its breaking down, hazardous to handle/breathe?
If it has asbestos in it, any dust is very[sup]*[/sup] hazardous. You should not handle it, and should take careful steps to safely dispose of it. Such steps will almost certainly be regulated, so you should find out what you local options are. Going into the ordinary trash is almost certainly illegal.
In general, such materials are not hazardous when they are not breaking down - meaning you have more safe options to dispose of them. But there are zero reasons that you want to have this stuff about.
Asbestos is nasty. What is more, the foamy asbestos is nastier than other types, such as asbestos embedded in cement panels, because it sheds more easily the microscopic fibres that, when inhaled, get stuck in your lung and cause all the trouble. The best way would be to disturb it as little as possible while you have it around, i.e. don’t shake it or scratch it, and to get rid of it safely.