iPhone 5s home button is pushed in too far, is it fixable?

Take it to an Apple store genius bar. Some Apple stores in the US now are able to do repairs on the iPhone 5-series in-store. Those include the screen, home button, and power button.

If they cannot, they will discuss a replacement with you.

Those stores that cannot do the repairs use the replacement of the entire iPhone approach. Either way, be sure and make a backup before going in, and in the iCloud if possible as they can restore from an iCloud backup if things get wiped.