If you have another device, you might try signing into your gmail there and testing to be sure it's not just a problem on the iOS device and it needs to be restored from a fresh OS (or out of space on the filesystem). Since iOS is designed to not be like a computer and pop up all sorts of error messages, sometimes when an error condition has arisen, it's not obvious why things don't work. In fact, you might just need to power down the phone and let it reboot if it's a temporary issue.