I am using Google Hangouts via Gmail on a desktop web browser, and I want to send a text message to myself. Is it possible? And if yes, how?
You can achieve this by having a named group chat with no other members.
Start a new conversation, choose New Group, and enter a name for the group. After typing a name, you can start the group without having to add any people to it.
You'll need to send a message before leaving this screen, or the group will not get added to your conversation list. Once you've sent a message in the group, you can continue to use this group to message yourself in the future, or you can leave the group to terminate it.
Thanks to Ben Plont and Vembu for their answers in the comments:
No it is not possible to text yourself in hangouts.
You can email to your cell number and it comes in as a text. Check this reference guide for the proper format for your provider. emailtextmessages.com for instance, If my cell number was (123)456-7890 and I used AT&T as my provider I would email firstname.lastname@example.org
From the Hangouts pane in Gmail, click on the phone icon (not the chat/text icon), then click the + icon near its top, then find yourself, then click SMS on yourself.