Tragically, only in a virtual reality website. I wonder if we can persuade them to let people for it for real – for charity of course.
However, while bungee jumping is not possible, people do occasionally abseil down the clock face to clean it – and occasionally climb up the outside to protest about something or other.
http://londonist.com/2009/01/bungee_jumping_off_big_ben.php

Although – if you want to visit Big Ben for real, you can go on tours of the Clock Tower and stand next to the infamous bell as it strikes the hours.
http://www.ianvisits.co.uk/blog/2008/11/11/visiting-big-ben/
