In case it helps anyone: That error also occurs when there is not enough RAM available (and Elasticsearch didn’t start because of this).
I installed Zammad on a virtual server with 2 GB RAM (here they mention that as a min., and why not try it: How to Install and Configure Zammad Ticketing System on Ubuntu |).
After letting Certbot change my Apache configuration I thought that may have messed up the http connection to Elasticsearch.
This helped to think about the RAM:
After rescaling the Server to 4 GB RAM it worked to start and connect Elasticsearch.
@MrGeneration
It may sound stupid, but maybe you could add a warning to the documentation (Set up Elasticsearch — Zammad System Documentation documentation) that Elasticsearch really needs 4 GB RAM to even start. Cause the error messages don’t mention the RAM problem. And people who just want to set up a test machine may try it with the smallest/cheapest virtual server.