Is there a physical fix?
Part of the price tag on your iOS device includes access to the help you need to keep it in tip-top shape. Once you get it into the store, the staff will take care of you -- just know that most fixes, unless you've paid for extra insurance, will come at a cost. If you've ruled out software issues with the troubleshooting steps above, most especially plugging it into your Mac or iTunes and seeing what happens, you're most likely looking at a hardware problem.
Faulty home buttons are a legitimate issue, and if that's what you've got, most experts would suggest simply replacing the button itself. Apple can do that for you, or if you're feeling confident you can even do it yourself, following along with online guides.
But Apple doesn't recommend it -- there are some compounds that can mess up the screen, so be wary -- but if you're feeling OK with your skills, some DIY experts recommend this: just a couple drops of nail polish remover on the Home button itself, after turning off the phone. Click the button a few dozen times, wipe off the remaining liquid, give it a half-hour, and switch it back on. If you're brave, we hear this could be the winner.
If you're still experiencing trouble -- and haven't accidentally damaged your phone! -- you're really at the end of the line. Call or contact the helpful Apple support team. And good luck!
If you find yourself stuck in an app and can't go Home, simply turn the device off and on and you'll boot to the Home screen. Then, under Settings > General > Accessibility > Physical & Motor, you'll find the Assistive Touch setting, which puts a virtual Home button directly on your screen. It's engineered for convenience, which means it'll follow you into any App and you can move it around as you like. Now you can just use this button and head to the AssistiveTouch menu. Once you've got the option enabled, it'll be there for you regardless of how your Home button is acting today.