In a data storage apparatus for storing data by making the data to be a file, a file is deleted, in a normal deletion processing, after contents of the file are destructed by overwriting the file to be deleted with data for elimination such as "all-zeros". On the other hand, with respect to storage processing, when storing new data, files waiting for deletion are investigated whether they are existent or not, excluding files being deleted currently in the normal deletion processing, and when files waiting for deletion are existent, either one of the files waiting for deletion is overwritten with new data to be stored.