Update: The issue (#10230007) has been marked as resolved by the Play Store team. Apps should install and update fine now.
For the last week or so, a great number of Android users have become increasingly frustrated with frequent "Package file is invalid" errors when updating or installing apps. A variety of supposed fixes have been put forward by the community, but have proven to be merely placebos. Now Google has confirmed there is an issue. So no, it's not just you.
The severity of the problem seems to ebb and flow randomly. Sometimes apps update and install fine, then an hour later almost everything fails. The package file errors may also be connected to "497 errors" reported by some folks.
The issue now has its own page where new information will be posted. For now, stay strong. We'll get through this... somehow.