2014 11” MacBook Air stuck in Internet Recovery

Those errors can indicate a networking issue. Try connecting the laptop directly to the router using an Ethernet Cable and an Apple Thunderbolt to Gigabit Ethernet Adapter for a faster more reliable connection that also bypasses hard to troubleshoot WiFi issues.

If you have access to another Mac capable of running macOS 10.11 to 11.4, then you can create a bootable macOS USB installer using the instructions in this Apple article:

https://support.apple.com/en-us/HT201372

The description sounds like the internal drive does not have any Empty Space for the OS to load and function. If this is correct - the computer will eventually stop booting at all - period.

Transfer files between two Mac computers using target disk mode and you may able to transfer some personal files off the drive and that may or may not allow the computer to boot up. Then, some serious house cleaning all be required. See used and available storage space on your Mac,

• Other: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.