In general: Yes.
Does it make sense? For most people not.
The Estonian Health Insurance Fund “Haigekassa” has a territorial scope - just like any other national universal healthcare system. So coverage only makes sense to you if you possess a European residency permit or citizenship. And if you are not already contributing to the social security of another European country.
In other words: Obtaining coverage through Haigekassa is an option to look into if you are living a nomadic lifestyle inside Europe and are not restricted to travel to Estonia at any given time. The reasons for that:
Through Haigekassa you are covered for all medically necessary treatments and medication provided by public hospitals and clinics in Estonia. In addition, you are provided with the blue "EHIC", the European Health Insurance Card. With this card in your pocket, you have access to the European network of social security agreements. This means that you are covered for medical emergencies all across the European Economic Area (EEA) - if treated in public hospitals and clinics. For every extended treatment that goes beyond short-term emergency treatments, you would have to travel to Estonia.
How do you achieve eligibility?
As Haigekassa is a social solidarity system there is - other than with private health insurance providers - no medical risk assessment at the beginning. Once you start contributions you are covered - even for all your pre-existing conditions. The contributions depend on your work income. As soon as your Estonian company pays a board member’s salary to yourself of at least the Estonian minimum wage - as of 2024 that's 820€ (725€ until 31.12.2023) per month - you will be covered by Haigekassa starting 14 days after your first payment.
Board members' salaries are fully taxable in Estonia. So this means on these (at least) 725€ per month you would pay 20 percent income tax (145€) and 33 percent social tax (239.25€). As long as you keep paying this monthly salary including the contributions you are eligible for Estonian social security.
This answer is provided by XOLO client Christoph Huebner. With his company NOMADS.insure he provides monthly webinars on the options for global health insurance for e-Residents of Estonia.