You may have already tried this, but have you downloaded it to a local machine (MAMP setup) or other hosting to confirm that it is indeed an issue with your specific host? I’m not discounting that it’s an ACF issue, but it’s at least something to try so you can gather the most info possible about what is going wrong.
I wanted to update everyone in this thread on a solution I found to the activation error problem. Obviously it might not apply in every case, but it’s worth trying to make sure it’s not easily fixable.
I have the same activate issue described in several posts (before trying to activate and another error after trying to activate). This was on a site that was previously activated and functioning fine. Upon further inspection, I also noticed that my Gravity forms Developer license would not activate either (similar symptoms). This let me know it most likely wasn’t only an ACF issue, but something was effecting both plugin activation methods.
My Solution:
After about 2 hours of trying to figure out a solution, I ended up downloading the newest core files from wordpress.org and re-uploading them to the site (be careful not to overwrite your wp-content folder). This instantly fixed the issue for both plugins. I made no other changes to the site files. It was the same version of WP that was installed already (so no a version issue).
My only guess is that there was some sort of error in the core WP files from an incomplete WP update that was effecting the activation calls. I wasn’t getting any PHP errors, so I can’t pin down exactly what was happening.
If you’re having issues, I would suggest trying a re-install of the core files for a quick attempt at fixing the problem. Hopefully it will help some of you.
