I'm applying to work as a machinist, but as student just leaving technical school I'm still somewhat green. For this reason I don't expect to be given the paid position, but I would still be interested in doing unpaid work for the sake of work experience. Would it look bad to mention my willingness to work for free when applying for a paid job?
Would it look bad to mention my willingness to work for free when applying for a paid job?
"Bad" I don't think so, but as Joe said in comments it would be unusual, as part of applying to paid positions is, well, because you want to be paid.
Given that you doubt you may land that (paid) job, but still would not mind working pro bono, what you can do is apply to the job, if you get it great, if you don't then ask if they have any unpaid internships or similar positions available.
This will still give you chance to see if you do land the job before offering your free services. Eventually you will have the confidence and experience needed to negotiate or land jobs with higher pay grades.