Xbox – Can’t sign into profile to save in GTA 5

Go into your system settings and select storage, then select gamer profiles. If any of these are corrupted then you must delete the profile only and not the data then redownload those profiles onto your xbox. If this is not the case then you might need to delete your system cache: http://support.xbox.com/en-US/xbox-360/settings-and-initial-setup/clear-system-cache

If it still doesn't work then try these steps provided by XBox Live support

1. On your console, go to Settings, and then select System.
2. Select Storage, and then select your primary storage device.
3. Select System Items.
4. Select Marketplace System Data.
5. Select Delete.
6. Restart your console.

If that still doesn't fix the issue then it is most likely a networking issue which you should then try hooking up a laptop or another computer to the same connection in your XBox, as in physically unplug the cord from the XBox and plug it into something else that can use the internet to test if the connection is actually good or not. If it is a wireless connection then you might have to manually enter your network preferences in the XBox Live connection options. For more help on that just contact Xbox Live support and have them guide you through the steps.

I have had many unfortunate times with XBox Live way too many times that I can count. Hope this helps!

It'll be because the saved game is strictly for your profile (your gamertag), you won't need the internet to save, but you first need to download your profile to continue the saved game.

Having your cloud enabled can also cause this problem - try disabling it in Storage settings.