Usually insurance for the house will be included in the lease, but it is for you to insure the furniture and other personal belongings. The insurance should cover events like: fire, lightning, storm damage, earthquake, explosion, riot and strike, bursting or overflowing water tanks, and theft.
Medical insurances are often covered as part of your contract with the organisation your are working with. You should however be familiar with the coverage of your insurance, and still carry with you ID and credit-card, so that you are able to pay on arriving e.g. at the hospital emergency ward.
As described in more detail under car insurances, it is highly recommended to have full coverage on motor-vehicles. Roads are rough, traffic is chaotic and damage to the care is not unlikely. Another type of insurance is getting a good 24-h guard service on your premises.