You can insure virtually anything - there will usually be someone around willing to take on the risk. However, we normally take out cover to protect ourselves and our family from losing out financially should the unexpected happen.
There are all sorts of policies out there helping you to protect your income or borrowing should you become ill and be unable to work, or your health if you want to skip the NHS queue or your possessions should they get lost or stolen.
You can get compensation for the unexpected costs that may arise when you are travelling, help with vets' bills and cover for your car if you crash it.