MacBook Air -2003F

It depends on the speed of your wifi and the speed of the internet connection.

Internet recovery will download about 600MB-1GB for the initial installer operating system. When you start installing it will also download the OS installer files that are around 6GB.

All of that can take a long time if the internet or wifi is slow (I have heard tales of 12 hours and longer).

If you have another Mac it is far better to download the installer & make a USB installer less things can time out or fail. It can be quicker if the wifi & internet connection are stable.

Try normal recovery mode first - hold cmd+r at startup. You appear to be using internet recovery (cmd+alt+r).

I think it is a networking error. If you have an suitable dongle please connect the Mac via ethernet & try again.

Another option is to create a bootable USB installer…

Create a bootable installer for macOS - Apple Support

Or use diskmakerx.com to create one.

If you cannot do any of those try moving to another wifi connection. The wifi that internet recovery uses has to meet certain criteria if I recall correctly.

Sign into the App store.

Download the macOS/ OS X version you want to install onto the Air. It may be listed in your purchases if you need an older version.

When the download completes quit the installer - you do not want to install it on that Mac.

Now follow the Apple's guide I posted above or use diskmakerx.com - it is a simpler process to use that app. It will copy the installer files to the USB disk.

Insert the USB disk in to the Macbook Air.

Boot it whilst holding 'alt'.

Select the orange USB disk to boot from & then run through the installer as normal.

NOTE: It can be tricky to get certain older versions form the App store if the Macs are different ages, post model info & the versions of OS X you want to install if that applies to you.