Trust me, if you have stuff like Gmail, YouTube, and Google Play pre-installed, you have Google Services Framework. It's there but it's NOT visible in your app drawer.
I don't know what kind of phone you have, but this is a rough guide on how to do what you're trying to do. Go to your phone settings. Under there, probably under a General or Other tab, look for Apps. You might have to look around for it but when you find it, you'll see that there are more tabs. Chances are the first one is "downloaded". Next will be "On SD Card" (but only if your phone has an SD card slot - if not, it won't be there), After that will be "Running", and next will be the tab that you want called "All"
Scroll down until you find "Google Play Services". tap it, scroll down until you see "Clear Cache". Hit the back button once, and tap "Google Services Framework". Tap "Force Stop", scroll down and tap Clear Cache. Hit your home button, and then reboot your phone. Hope this helps.