After Effects 2021 can't save my project file

I experienced the same thing until I switched from Team project to personal project under the 'Edit' drop down menu

So i experienced this issue today and i think i found a way to troubleshoot the problem. The easiest way is to export your project to media encoder then rename the temporary file that media encoder saves when you send a project to it. That file is usually a replica of your original project.
So you can close your after effects, launch it again, open the file and continue working on your project.

I hope you find this helpful even if it's 2 years after your post

Okay, I think I fixed this in some weird way and I'll try to explain incase it helps someone else.

After trying all these suggestions (going through finder to change read/write settings, finding a security/privacy setting, collecting dependencies) I eventually tried to see if I could somehow export the entire project file through Media Encoder (since Media Encoder is allowed to Write onto my computer). Once I clicked on the link that lets you choose where it exports, I noticed that it wanted to save next to a AE file that's name ended with the title of my comp. So I individually sent each comp to media encoder until I had created a bunch of AE files in some random generated location. Then I opened the project folder in Finder and copied all the AE files into there. Then I closed everything and opened the main comp's AE file, and imported the rest of the files into the opened project. 

Now, I'm glad I got this far because I know my project saved. But I still have to figure out how to make it save from AE, so I don't have to keep using Media Encoder. 

Hope this makes sense. If not, I'll try to recreate it and screen record the process. 

It appears that this has now been fixed in 22.6!

Like with all other Adobe apps,  After Effects will prompt you to connect to Adobe's servers every 30 days - which is far better than having to connect every 3 minutes. If you are still having problems, you can use offline activation to generate a response code.

i change dns to google dns 8.8.8.8 and 8.8.4.4 its work for me. 

After what seems like months and months of problem solving, I've (hopefully) deduced it's a PlusNet DNS Server setting issue that stops Creative Cloud connecting with the knock on effect of screwing up After Effects. Fingers crossed this seeeeeeeemmmmmss to work!!!

https://community.adobe.com/t5/download-install-discussions/plusnet-authentication-problem/m-p/12976730#M553377

Possible solution! After another 2 hours wasted this morning not being able to Save or Import anything. Have definitely narrowed it down to Creative Cloud not connecting to Adobe's server. Messgae popping up "Unable to reach Adobe servers" I was unable to even manually log in on their website.

Then found this (fingers crossed solution) that may help future generations searcing endlessly for answers. I just clicked on the two links it took a few moments to connect so be patient then all of a sudden I'm back up and running. Can't even pretend to understand it but I hope it works for you too.
https://helpx.adobe.com/uk/x-productkb/policy-pricing/activation-network-issues.html
Why someone from Adobe couldn't have suggested this months ago I don't know.

I spent a few hours today trying to get to the bottom of the issue and it appears that After Effects currently REQUIRES an active internet connection in order to save projects locally. 

After Effects would stop importing, exporting, saving and autosaving after 3 minutes when it couldn't connect to the Adobe servers. When my machine had internet access and I turned my firewall turned off, After Effects was fully functional for at least an hour.

If you have a firewall (LuLu, Little Snitch, Hands Off or the default MacOS firewall) then try disabling it before launching After Effects. If your machine is unable to connect to the internet then there's no solution until Adobe fixes this bug.