You clearly not followed the automated installation for ubuntu and setting 777 to data directory you are opening the security vulnerability for your website. Too many redirects that means you have tried to do something else with your installation.

Simply use the installer https://www.opensource-socialnetwork.org/wiki/view/3994/how-to-install-ossn-on-ubuntu-1804-2004-x64-using-command-line without editing any files or touching any services. Once done simply follow the steps in UI.

I used the command and managed to get rid of the message:
chmod -R 777 /var/www/ossn_data
But when I complete the data and give it further it brings up the message:

ERRTOOMANY_REDIRECTS

Well I used the command:
chown -R www-data:www-data /var/www/html
and it fixed my permissions problem. After typing https:// example.com/verifyperms.php, it no longer shows me errors regarding those permissions.
And that first error with MOD_REWRITE remains. I bypassed it, but another error appeared:
"Invalid data directory or directory is not writeable."
What should I do?

I used this tool and it showed me the problem:
Permission problem found!
The owner of /var/www/html is: root - please change to: www-data
How can I solve it?

Following the instructions in
https://www.opensource-socialnetwork.org/wiki/view/1101/how-to-report-a-bug-error-issue
Case A
might be of help

I uploaded the files to /var/www/html, but the installation still won't start. All the time error with permissions.

I have used these commands, but the question is, where do I upload the files again now?

https://www.opensource-socialnetwork.org/wiki/view/706/how-to-install-open-source-social-network

Why don't you simply use https://www.opensource-socialnetwork.org/wiki/view/3994/how-to-install-ossn-on-ubuntu-1804-2004-x64-using-command-line

It worked, but now this error has popped up: "Invalid data directory or directory is not writeable."
I know it is about entitlements. How do I set them up so that everything is secure?

Then mod rewrite is working there is other problem that has been addressed in next version but you can bypass this step now as it shows 1, change ?page=settings to see the settings page instead of checklist.