If you are not employed in France, and therefore do not have an employment contract, you need to distinguish whether you are registered as a student in France or not.
If you are enrolled in a higher education establishment in France, you can benefit from the Protection Universelle Maladie (Puma), but you will need to apply for affiliation and a social security number in order to be covered.
If you are NOT enrolled in an institution of higher education in France, you can benefit from the Protection Universelle Maladie (Puma), but only after 3 months in France and provided your stay exceeds 6 months. You will also need to apply for affiliation and a social security number in order to be covered.
In both cases, you must be covered by private health insurance until you are registered with the social security system.
If you are NOT enrolled in a French higher education establishment and are staying in France for less than 6 months, you will not be eligible for the Protection Universelle Maladie (Puma) and will need to be covered by private health insurance (sickness, hospitalization and repatriation).