Make sure Chrome is allowed access to the folder where you are downloading.
Control access to files and folders on Mac - Apple Support
Are you using any browser extensions or other third party software which may interfere with the normal operation of macOS? Typical culprits are anti-virus apps, cleaning apps, and third party security software....none of which are needed on a Mac as they usually cause more problems than they solve. You can try booting into Safe Mode to see if it makes any difference since Safe Mode prevents third party apps from launching automatically during boot & login. You can also try creating a new macOS user account, then logging out of your main user account followed by logging into the new user account to see how things work. If things work in the new user account, then you have some sort of configuration issue with the main user account or you may have an issue with a login item or browser extension.
You can also try using another browser to see if it also has the same issue which may help provide more clues. Plus if another browser works, then at least you can save your downloads.
Do you by any chance have iCloud enabled?
Also, where are you trying to save the downloads? FYI, You can only save files to your own home user folder.
Edit: Sometimes just rebooting a computer can clear up problems.