Is there a way to automatically run
get_header() at the beginning of template and
get_footer() at the end of it?
Currently my code is too repetitive with those calls.
I've been looking for a way to do this, but I can't find it in Google.
You could hook into that filter, handle the including in a callback function and return
Sample code, not tested:
@scribu proposes a method for reducing code repetition in WordPress templates which he calls theme wrappers, which achieves what you're looking for. The roots theme uses a slightly modified version of this method.