Asbestos is a natural fiberous material and doesn't rot, but the insulation it is contained in may be deteriorating and coming apart. That's not good. The fibers have been proven to cause lung cancer, but it has to get into your lungs first. You can have it removed, probably not something you should do yourself. Or, encapsulate and/or cover it so the fibers don't flake off into the air.
If you do get sick, you may be on your own. The trial lawyers have bancrupted just about everybody connected with asbestos production and manufacture. They are going after the little guys now, so called 3rd and 4th layer defendants. Over $50 Billion has been paid out (about a third of that to attorneys!). It is estimated that about another $100 - $200 Billion (yes, that is billion with a B) will be paid out by companies and their insurers before it is all over.
(my uncle was a steam fitter on ships and died after horrible suffering from asbestosis, they used asbestos insulation on the pipes)