There is a reciprocal healthcare agreement in place between the UK and Estonia, which guarantees free provision of basic emergency treatment on production of a valid E111 form (available from post offices).
E111 is not a substitute for travel insurance, and you should still take out a comprehensive policy prior to travel in case you need to be evacuated back to the UK in an emergency. Prior to being allowed into Estonia, you need to prove that you have private medical insurance up to the value of 400,000 Estonian Kroons (approximately £18,000) - it is important to check your policy and the exchange rate prior to travel.
Tap water is considered safe although many people stick to bottled water in the country for preference.