Macbook air stuck during OS upgrade

Have you tried pressing and holding the Power button to shutdown the computer, then wait one minute and restart. Do you boot into Snow Leopard? If you are unable to start, then hope that you have a Recovery HD from the failed El Capitan install, and do this:

Reinstall El Capitan Without Erasing Drive

Please be sure you back up.

1. Restart the computer. Immediately after the chime hold down the Command and R keys until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the the side list.
4. Click on the First Aid icon in the toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Reinstall OS X and click on the Continue button.

Also, see this User Tip: Basic Steps for Upgrading OS X.

[Permission to use part or all of the above has been granted by Kappy, exclusively, to theratter.]

If the disk cannot be repaired then it may have failed. If you use your other computer via Target Disk Mode, then any Firewire will do although FW800 would be preferred, but both machines need to support it.

It's unusual for an SSD to fail, but they can even if rarely. You may need to try the following:

Snow Leopard Erase and Install

1. Insert Snow Leopard DVD or the original installer disc that came with the computer when it was new into the optical drive and restart the computer.
2. Immediately upon hearing the chime hold down the C key.
3. Release the key when the Apple logo appears and wait for the loading to finish.
4. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities menu. After Disk Utility loads select the out-dented disk entry from the side list (mfgr.'s ID and drive size.) Click on the Partition tab in the Disk Utility’s right window. Set the number of partitions to (1) from the drop down menu. Click on Options button and select GUID, click on OK, set the format type to MacOS Extended, Journaled. Finally, click on the Apply button.
5. After formatting has finished quit Disk Utility. Continue with the OS X installation and follow the directions.
6. When the installation has finished the computer will restart into the Setup Assistant. After you complete the Setup Assistant you will be running a fresh installation of Snow Leopard. Open Software Update and install the recommended updates.
7. If you are planning to sell or give your computer away, then do the following: After you reformat your hard drive and reinstall Snow Leopard, the computer restarts to a Welcome screen. To leave the Mac in an OOTB (Out Of The Box) state, don't continue with the setup. Instead, press Command and Q keys to shut down the computer. The Setup Assistant will guide the new owner through the setup process after he/she turns on the computer.
8. Download and install Mac OS X 10.6.8 Update Combo v1.1. if needed. It is not needed if you are selling or giving away the computer.

If you are not able to do the above because of a failed disk, then the SSD may well have failed.

[Permission to use part or all of the above has been granted by Kappy, exclusively, to theratter.]