It was a common practice to use materials that contained asbestos in them in home construction prior to the mid- to late 1970s. There are very specific guidelines for asbestos removal, however, according to the EPA, it is also perfectly safe to encapsulate the asbestos.
For flooring, this means you can install directly over the old material as long as you don’t disturb it. Ceramic or porcelain tile, stone, or a floating floor of any type would qualify. By encapsulating the asbestos without disturbing it, you have effectively removed any hazard.
You can also pour a floor leveling compound on top of the old material in case the floor has dips and divets. The key to safety is simply not to disturb it.