If you are taken on as an employee by someone then it is their responsibility, as an employer, to ensure that they have Employers' Liability insurance. Their policy must have cover for at least £5 million and come from an authorised insurer.
It is important that you ensure that your employer has such a policy as this will protect you if you damage anything or cause an accident or injury to someone else whilst working for them.
It will also provide cover for your employer if anything should happen to you whilst at work such as injuring your back whilst carrying out a work task.
If you are self-employed, then you need to have your own Public Liability insurance policy in place to protect you whilst you are working for someone. Because you are self-employed, it is important that you are protected in case you cause any damage or break something or if an accident occurred as a result of something you did. For example, if you were pushing someone in a wheelchair and you accidentally knocked in to another person and injured them, that person could claim against you rather than the person who has engaged you.
Your employer should have some form of home contents insurance in place to protect their belongings In the event of an accident. It is not your responsibility to have such insurance for when you are working in their home.