The following information comes directly from the Zotero documentation:
We strongly recommend that you regularly back up your Zotero library. Syncing is not a good substitute for backing up: the Zotero servers only store the most recent version of your library, and it takes just a single (possibly automatic) sync to change the server copy.
Rather than backing up just your Zotero library, we recommend using a backup utility that automatically backs up your entire hard drive to an external device on a regular basis. Most modern operating systems offer such functionality (e.g., Time Machine on Macs).
If you want to back up your Zotero data specifically, locate your Zotero data, close Zotero, and copy your data directory (the entire directory, including
storage and the other subdirectories) to a backup location, preferably on another storage device. As with all important data, it is a good idea to back up your Zotero data frequently.
Warning: Zotero lets you export your Zotero library as a Zotero RDF file. However, exporting and importing your library via RDF won't result in an exact copy of your library, and it isn't recommended as a backup strategy.
For step-by-step, illustrated backup instructions, see this guide provided by the Northwestern University library.
You can buy more storage beyond the free 300MB limit.