In my 15 years running two companies and two bands, I've employed about 100 people.
So from an employer's point of view, here's my best advice on how to get hired:
Focus on one company
Do some soul-searching to decide what you really want to do.
Then find the company in your area that you feel is doing it the best. (The company needs to be near you already. If you don't live near them, move there first, or choose a closer company. Do not do this remotely.)
It doesn't matter if they're not hiring.
Learn all about them. Read every page of their website. Become a customer. Read every article about them. Study and memorize this info.
(This only takes a few hours, and is a much better use of your time than blasting resumes.)
If you don't really want to work for this company, pick a different company and do this section again.
Tell them how much you want to work for them
Start contacting them to tell them how much you want to work for them.
It doesn't matter who you speak to first. Start with anyone. Just start.
Tell them, (in your own way), “You are my favorite company. It's my dream to work for you. If you have any aspect that could use a little help, let me do it, and I promise you it'll thrive. I'm that passionate about this.”
Eventually, contact different people in the company, especially the executives, not just human resources.
Ideally, you could be more specific, telling them ways you could improve one of their projects, services, or products.
Be persistent (though succinct)
Combine phone, email, and in-person. You must use all three methods, since each has its strengths.
Always be succinct. Don't take more than two minutes of their time. But always show your passion, and how much you can help them.
Vary your message. Sometimes ask advice. Sometimes give advice. But always make it clear how much you want to work there.
Do this every week. It's OK to be almost annoying. Polite manners don't prove passion.
Do this until hired
Eventually they will be hiring, and they'd be damn foolish not to hire you.
Especially when faced with the alternative of opening up the floodgates to help-wanted ads, they'll much rather go with this person who has persistently proven their passion.
(Could do this with a few companies at once)
If there are sincerely a few different companies you would love to work for, and you have the time, consider doing this process for a few companies at once.
P.S. For further inspiration, read how Tom Williams got hired by Apple at 14, using this method.