To change your status setting in Google Chat, click on the down arrow next to the current status indicator at the top of the Chat screen. In this case it shows a current status of "Active":

NOTE: If your Chat window is not full screen, you may just see the "indicator dot", without the status description.

When you click on the down arrow, you will get a menu to select a new status or to create a custom status. Select the status option to match your desired status.

NOTE: Be sure to make sure you don't inadvertently leave your stats set to "Do Not Disturb" or "Away" so that you don't miss important notifications.