Amanda Lightstone's reply:
I appreciate that you are finding it frustrating to get into the role or organisation you want, especially when you are appropriately qualified for the job. As you will know all too well, it is a competitive job market out there. Therefore, anything you can do to set yourself apart from the other candidates will help.
Make yourself known by volunteering within the organisation you wish to work. It need not be a full-time volunteer role, but even helping out at education or outreach events will bring you into contact you both with the public (enhancing your skills) and the staff (who could be your future employer). In addition, I encourage you to adapt your CV or covering letter specifically for the role to which to are applying. Even a little bit of research into the organisation's current projects or activity will shine through in your application.
Keep striving for your ideal job and stay true to your passion.