Macbook air 2013 not showing internal hdd

Did you replace the original Apple OEM SSD with a third party SSD? If so, then you need to boot into macOS 10.13+ before the third party NVMe SSD can be seen. Try booting into Internet Recovery Mode using Command + Option + R.

If you never changed the internal SSD, then your internal SSD has failed since it does not show up on the left pane of Disk Utility.

If the internal SSD was previously replaced with a third party SSD and you do not see the internal SSD on the left pane of Disk Utility while booted into macOS 10.13+, then that third party SSD or its SSD adapter has failed. While booted into macOS 10.13 installer, you need to click "View" within Disk Utility and select "Show All Devices" before the physical drives appear on the left pane of Disk Utility (macOS 10.13+).

If you purchased or acquired this laptop as used/pre-owned, then perhaps the previous owner replaced/upgraded the SSD to a third party SSD. Booting into Internet Recovery Mode using Command + Option + R to attempt to access the macOS Big Sur online installer is an option just to make sure. Unfortunately some Macs may only boot to the online installer for the older OS which originally shipped with the Mac from the factory.....which is too old if the laptop has a third party internal SSD installed.

If it is an original Apple OEM SSD, then booting to a later version of macOS won't make a difference since the physical internal SSD is not showing up in your picture of Disk Utility. I guess you could try reseating the internal SSD just on the off chance the contacts are oxidized or the SSD has become slightly dislodged. You will need a Pentalobe 5 driver to remove the Bottom Case, followed by disconnecting the Battery from the Logic Board before you use a Torx T5 driver to remove the internal SSD.

If you see any liquid residue anywhere inside the laptop, or you see any corrosion (looks like bluish/greenish white stuff) on any of the internal electronics, then it indicates liquid damage to the laptop. If that is the case, then as long as the SSD's contacts are not damaged, then perhaps the SSD may have survived since most of the SSD's electronics except for the contacts are covered in an resin that can protect the SSD.

FYI, the original OEM Apple SSD was known to fail in some of the 2013 MBPros (Apple had a free repair program at one point which has long ago expired). The SSD is the weakest link assuming the laptop has not suffered any accidental liquid damage.

Sorry for the bad news regarding the SSD. I hope you have a good backup, otherwise you will need to contact a professional data recovery service to see if they can recover any data from the SSD.

Apparently you have erased (deleted) the computer's operating system. That means your data is also deleted and gone unless you are using a backup app such as Time Machine or Carbon Copy Cloner. You need to begin by reinstalling a new OS to get the computer working.