If running the synchronization cron returns an error, always check your server logs to understand the reason behind that.
If a server-side timeout problem occurs, keep in mind that on large catalogs synchronization may take a VERY long time.
You can check the number of products to be processed by cron by consulting the product lists in the module’s Sync TAB (add / update / close products) and estimate a processing time of about 3 seconds per product, useful to process the request / response communication with eBay.
If this time is longer than the maximum execution time of your server (the value is visible in the “System Information” screen of the module), the cron could generate a timeout (the module sets a directive to increase the maximum execution time to 3600 seconds, but this setting could be ignored by the server depending on your hosting configurations).
As already indicated in the documentation, the cron should NOT be used as a substitute for a massive synchronization of products or large catalogs, which should always be done through the module using Fast Sync functionality.
The product synchronization cron is designed for those who use external import systems and need quick and frequent updates of products that have modifications (see the cron section of the documentation for more details). You won’t need any cron if you use Prestashop in a standard way (via backoffice) because the module will automatically synchronize the products when you save them through the hook system integrated in Prestashop (see “How the module keeps products synchronized” in the FAQ for more information).
If you really need to update products by cron and have a large catalogue, keep in mind that Fastbay offers the “reviselastimport” cron parameter , that will only detect products with a recent modification date than the last eBay synchronization. This would allow you to use this kind of cron to make sure you are going to update only really updated products.