Is there a way to change your status in Google Hangouts from within Gmail?
Actually it's possible to become invisible to some extent. Quoting Change my status and profile photo:
Invisible: "Invisible status" isn't available in Hangouts, but you can use it by reverting to Google Chat. When you set your Google Chat status to invisible, Hangouts users will see you as unavailable.
Tip: If you're busy and you don't want to be notified of text messages or calls, you can mute notifications. People who send you messages or invite you to video calls won't be notified that you muted notifications. But they'll see your profile picture without a green circle, indicating that you might not be immediately available.