I would like to send a text message to an e-mail address using Messaging on Samsung S4. I have already done it before as a test, but I cannot get it to do it any longer.

How to enable this feature again?

You have to enable MMS, (it's usually already enabled), and when you have done so, you can turn 4G on. Once that is done, and 4G/LTE, 3G is on, then you can type an email to send to.. type your message, and voila.

How To Send a Text to an Email Address

I hope this helps you. It is a test that will help you determine if you can send SMS to email. Open your regular text messaging app and type a text like you normally would, but in the 'To', field enter your email address. Send one to yourself first to see if it works.

There is also a carrier specific ones can use. For example My phone is a Boost phone:

1. SMS 12345678912@sms.myboostmobile.com
2. MMS 12345678912@myboostmobile.com

Here's a more general answer for someone who's looking to do this from any carrier. There are apps that can do such a thing. If you're looking to just send future texts straight to email, there are apps like SMS to Phone/Mail. For those who want to backup any SMS/MMS from the past and either print, convert to PDF, or email you can use an app like SMS Backup Print & Restore. The second one here has a pro version as well.

Alternatively, you can automate this process through apps like Automate or Tasker. Those are definitely for more advanced users though.