Natural alternatives to synthetic
Luckily some alternatives to plastic floss exist. The two main options are bio-based plastic and silk floss. The former biodegrades in an industrial compost the latter biodegrades in your backyard compost.
A reusable floss dispenser
To be really zero-waste floss, the container should be reusable as well. A reusable floss dispenser can be refilled with floss coils. Reusable floss dispensers are made of stainless steel.