One thing that is attractive about computer programming or web development as a career is that you can acquire many of the skills that you need to do it professionally without first convincing someone to hire you so that you can get work experience. Entry level software jobs are something that you can create yourself.
Creating and executing your own projects is a way to display your talents for any future employer. Obviously breaking into the industry still won’t be easy if your work is of poor quality, but having no formal work experience doesn’t have to be a strict barrier to entry, as it would be in many industries.
Lou Franco over at Atalasoft has an excellent list of seven ways to get software work experience without having to actually be hired by anyone.