Government service is a fine career. Go for it!
Your cover letter with your application should briefly say who you are and why you are interested in government service. Be sure to mention "serve" or "service"
Use a little imagination (but not too much) when you look at the title and department of the internship. Say something about how that internship will help you learn more about service.
For example, "I am Naysa B.., a rising senior at Fort Washington High School. I'm hoping for a career in government. I've served as a library assistant (or whatever) in my high school, and I enjoy helping my fellow students find their books (or whatever). Your internship in the Department of Education (or whatever) is an opportunity for me to learn more about government service by getting experience with the work of your office. Please call me at 234 456 4444 to let me know when I can come to your office. I look forward to meeting you and hope to work for you.
See the formula?
one kind of work you've done,
what it taught you about your desire to serve
the name of the department you're applying to
what you hope the opportunity can do for you
an offer to work.
Wrap up your letter with a call to action (call me at xxx) and repeat the offer to work.
What you've already done is important, even if it doesn't seem like it. "I've served on the crew at Burger King. My favorite part of that job was greeting customers to take their orders." would be excellent (if that's what you have done.) it proves you know something about working with the public. It also show you are able to take something good from each experience you have. You may have some war stories from your experience ("my least favorite part of the job was cleaning the grease trap, so I learned to do it right the first time"). Save those for the interview if they ask about stuff you don't like doing.
If everything they need to know to get you on board is in the cover letter, that helps. Make your application the easiest one to act on. They're hiring an intern so, honestly, your resume is not a make-or-break deal. If they don't have to dig into it carefully, you're saving them time.
A serious piece of advice for somebody interested in government. Call your state legislator's office, tell the constituent service person you're interested in a government internship, and ask for suggestions about good internships for you. Believe me, they LOVE getting this kind of call. If you have an elected official's endorsement, you will get the internship, and you may also get a helpful mentor.