Hi Hojiaka. Congrats on your new Moodle admin role!
Exporting to the various formats can be a demanding process on the server. It is common for an export to cause PHP execution to reach the various limits and controls that are often set at default values. These limits can exist in the PHP configuration, the web server configuration, and many other layers depending on your hosting setup (such as load balancers).
I would go out on a limb and take a guess that you are either hitting a time limit (in PHP or web server such as Apache) or a memory limit in PHP.
When you say you are trying a .xle extension, you are referring to an Excel file, is that right?
Do you know about how many records are in the report?
Does it work if you try to download as a CSV (plain text file)?
I would approach this issue by reviewing the web server error logs and the PHP error logs. It would be helpful to also enable debugging within Moodle while attempting an export. Any debug errors might tell you at which layer the process is failing (web server, PHP, or neither). You could share any debugging output you receive.
Once you can identify more diagnostics, it may then be easier to understand if PHP or the web server configuration needs to be tuned. You might also want to review your current settings in php.ini for max memory and max execution time.
Jamie