I've seen this question asked many times and was once in your situation i.e average grades from a state school, little to no experience, etc...
Several years ago I took an approach similar to the following:
Find relevant contact names via industry associations, company websites, etc. For example, members of the
Bond Market Association. As a backup, go to the WSJ archives, etc and type in a word/phrase relevant to your career interests and scan the articles for relevant contacts...
Next step, find their address via their company's website and send them a letter via snail mail (email is too easy to delete/ignore). Something along the lines of
this (hopefully you can come up with something much better than this example) ending it with something like "I will call you next week to arrange a convenient time to speak with you..." DO NOT include your resume.
Before you call find out everything you can about their background, experience, etc (Google, NFA, NASDR, etc.)
When you call, will they offer you an interview/job? Probably not but many people are at least willing to give you 15 mins of their time (call after the market closes) to discuss their experience and your career interests.
It's not easy but it works if you do your due dilligence and are PERSISTENT.
For what it's worth, I know someone that recently used this approach to land a research position at a hedge fund and I've personally used it to obtain more than one position.