It is an interesting question. My instinct will quickly answer no because gzserver is built with a good bunch of shared libraries that needs to be loaded at runtime:

ldd /usr/bin/gzserver-8.2.0 | wc -l
258

Using cmake we could try to build gazebo libraries as static but that won't solve the problem of finding the rest of the libraries used by gazebo in their static (.a) versions. Build the whole stack of libraries that does not supply the static version from source can be ... eeh time consuming, being optimistic.

Seems like some people found your use case before and built some tools to generate static binaries (like http://statifier.sourceforge.net/). I've never used but sounds like a more reasonable approach to me. Please report your findings if you happen to go this way.

I am also interested in this topic. While looking at Gazebo source code, I notice there are files that shows example of compiling executable. Perhaps one can leverage this. Here is the link,

https://bitbucket.org/osrf/gazebo/src/bcbea7608b1ad88067ef652e8e7a4b93ec1001cf/examples/stand_alone/transporter/transporter.cc?at=default&fileviewer=file-view-default