The problems that have occurred are quite varied, the first thing to do is to update your server or ask the ISP to do it, especially cURL needs to be updated. This will resolve many issues with PayPal, eBay modules, and also with many modules that connect to social media like FB, even on platforms other than PrestaShop.
PayPal response:
SILOVER -> name=LIVE6.API.1&silo_version=880&app=appdispatcher_apit&TIME=2810113364; domain
TIMESTAMP -> 2015-02-25T07:49:59Z
For several days now, both the PrestaShop development team and PayPal have been working to resolve the issues. To understand what happened, we need to take a step back; at the end of 2014, a serious security flaw in the SSL protocol was discovered, and efforts were made to remedy it. On February 22, 2014, communication via the fsockopen protocol of the PayPal module for PrestaShop mysteriously stopped working, causing many users inconvenience and the inability to use it. Not everyone, because the module generally uses cURL, and only if this is not enabled does it fall back on fockopen. At the same time, since January, many WEB Servers have been experiencing an issue with updates related to the aforementioned security problem and the changes it brought; these problems are not obvious, so many do not pay attention, especially since there had been a recent intervention to update systems and avoid issues. Consequently, cURL works poorly, and eventually stops functioning.
Here for the solution: https://www.prestashop.com/forums/topic/419670-paypal-module-suddenly-stopped-working/#entry1982223