No, you can't handle an exception globally across all procedure/functions in a package.
The exception handler documentation says:
An exception handler processes a raised exception. Exception handlers appear in the exception-handling parts of anonymous blocks, subprograms, triggers, and packages.
Which makes it sound like you can; but the 'packages' reference there is referring to the initialisation section of the create package body
statement:

But that section "Initializes variables and does any other one-time setup steps", and is run once per session, when a function or procedure in the package is first invoked. Its exception handler doesn't do anything else.
If you really want similar behaviour then you can put that into its own (probably private) procedure and call that from the exception handler on each procedure/function. Which might save a bit of typing but is likely to mask what is really happening, if you're trying to log the errors, say. It's probably going to be simpler and better to have specific exception handling, even if that causes some repetition.