In Austria, as an employed person, you are insured against accidents under the statutory social insurance scheme, but this only covers damage caused by accidents at work or on the direct route between work and home, or by occupational disease.
Consequential damage from leisure and household accidents, on the other hand, does not fall within the scope of protection. If you are not (no longer) employed, you have just as little protection from the statutory accident insurance as children before the last (compulsory) kindergarten year.
Furthermore, kindergarten children, pupils and students are only insured on the direct way to the training center or against the consequences of accidents for the time they spend at these trainign centers.
With a private accident insurance you will have full protection without any gaps in the insurance coverage. A private accident insurance provides you additional payments, even if you already receive payments from another source, like statutory accident, health or pension insurance.