Can I sue if my job had me handle dangerous, cancerous material called aspestos tape?

In this day and age it sounds really unusual not only for your employer to have you handle asbestos, but for them to even HAVE it.  Also, just because something contains asbestos doesn't mean that it is dangerous.  As an example, Vinyl Asbestos Tile, contains asbestos, but it isn't friable.  You can sue if you actually have any damages from handling or being exposed to asbestos, but the way you say this is that you handled the tape and were sick soon after.  That is highly unlikely.  Asbestosis is a disease that builds up over time.  Perhaps I am reading too much into the way you wrote this, but the sickness sounds like it is unlikely to be related.  See your doctor.  Tell your doctor what happened.  Your doctor, not a lawyer, is the one who will be able to tell you if you are actually injured and have the right to recover either under workers compensation or as a personal injury. Having said that, if your employer is having you handle raw asbestos, that by itself is likely an OSHA violation.  Don't expect any recovery just because there is an OSHA violation, but OSHA will inspect and may fine your employer. For more information I have attached the OSHA link for Asbestos http://www.osha.gov/SLTC/asbestos/