Having acted as executor to a relative's estate last year, I am now very aware of the need to leave things in order for my daughters. Although most of my bills are online, I also have paper copies of most stuff, as I am not sure whether anyone else would be able to access my online accounts if I were to die or become incapable of managing my affairs.
My filing cabinet has most current documentation, e.g. insurance, car MOT, guarantees, receipts, and bank statements (paper copies mean I can scribble notes reminding myself what any large transactions are). I also have some box files with archived bank statements and details of closed savings accounts. Once or twice a year I remove and shred anything older than seven years.
Important certificates (birth, marriage, etc) are also in the filing cabinet. You need, ideally, to produce the birth certificate when registering a death, so it should be kept somewhere accessible (and not, as with my elderly relative, in a metal box fixed to the floor at the back the coat-cupboard!). A copy of my Will and a note saying where the original is stored are also there, and the Land Registry documents for the house. Some things need to be stored as original documents, even if you back them up by scanning to computer.