Is it linoleum or vinyl? How old is the material you are removing?

When mineral fiber asbestos backings were discontinued, they were replaced with fiberglass and other materials.

I highly doubt that your casual contact with asbestos, if you even had any, is going to be of any consequence.

Most important thing for you or for anyone else is not to smoke tobacco products or any other smoking product that introduces particulates and chemical compounds deep into the lungs.

To put this in perspective for you: Here is an excerpt from an article published by the American Society for Clinical Pathology that describes "the background level" of asbestos in ambient air.

"In regard to the first factor, studies indicate that everyone is exposed to background levels of asbestos in the ambient air. Studies have shown that members of the general (nonoccupationally exposed) population have tens of thousands to hundreds of thousands of asbestos fibers in each gram of dry lung tissue, which translates into millions of fibers and tens of thousands of asbestos bodies in every persons lungs."

Churg A. Nonneoplastic disease caused by asbestos. In: Churg A, Green FHY, eds. Pathology of Occupational Lung Disease. 2nd ed. Baltimore, MD: Williams & Wilkins; 1998:277-338.

Roggli VL, Pratt PC, Brody AR. Analysis of tissue mineral fiber content. In: Roggli VL, Greenberg SD, Pratt PC, eds. Pathology of Asbestos-Associated Diseases. Boston, MA: Little Brown; 1992:299-345.

You cannot eliminate your exposure to asbestos fibers anywhere on the planet, except maybe in industrial 'clean rooms'. However, being aware of situations that will introduce fibers in large quantities by repeated exposures is a prudent thing to do.