What you are asking for is called UMS or USB Mass Storage. On newer devices, this often got replaced by MTP (Media Transfer Protocol) for several good reasons. Both protocols have their pros and cons. Unfortunately, in most cases the user is not given the choice which one to use (some devices offer this).
However, an article at device-recovery.com explains How to connect Android devices to PC with USB Mass Storage Mode for different Android versions and devices, including your S4. I will list a basic exceprt here with no warranties: I have no S4, so I cannot verify this.
- Go to Settings→About Phone/Tablet.
- Go to “Build Number” at the end of the Scroll list.
- Tap on “Build Number” repeatedly (7 times). On your third tap you should see message indicating that you only have 4 more taps to go to ‘become a developer’. Keep tapping until you are a developer. (a toast message will tell you so)
- Now Developer Option will be included in setting list of yours now. Go to “Develop Options” to check USB debugging > OK. Done.
According to the article, that should have done the trick. I do not see how the two are connected, and doubt it a little. As an alternative, the very same article mentions a USB Mass Storage APK you could try as well.
For devices running Android 4.0/4.1, the steps are a little different, and sound convincing to me:
- Go to Settings→More
- Search then Tap on “USB utilities”
- Check the option: “Connect Storage to PC”
Unfortunately, my Optimus 4X running 4.0.3 doesn't show that option -- so again, I cannot guarantee this will work on any device.
A different approach is presented at this website -- again, explicitly for the S4:
- Attach your Samsung Galaxy S4 to your computer (PC or Mac) using the USB cable. If this is the first time you’ve connected your device to your computer, it will install the necessary drivers on your computer.
- On your Samsung device, touch Applications > Settings > Wireless and network > USB utilities.
- Remove the USB cable from you Samsung S4.
- On your Galaxy S4 touch “Connect storage to PC.” You will see a warning: “USB utilities / Connect USB cable to use mass storage.” Insert the USB cable into your Samsung device while the message is still on the screen. Do not touch “Cancel.”
- The screen will change to “USB mass storage” with a picture of a green Android and a message that says “USB connected.” Touch “Connect USB storage” at the bottom.
- Your Samsung Galaxy S4 is now connected to your computer. The device’s internal storage will appear on your computer as an external drive. If you also have a microSD card installed, this will show up as a second external drive.
Plenty other web ressources I've found on my search simply state: that's not possible without rooting. But it cannot hurt trying above methods: they might fail, but should not break anything.