Macintosh HD drive not showing using Disk Utility / Not able to reinstall Mac OX

It appears to me that the internal drive has failed and is not communicating with the OS at all.

The only thing Terminal sees is the disk image "OS X Base System". This is a required volume on every Mac, used in the background to facilitate booting, but it is not user accessible. Generally speaking that volume is to be ignored.

There should be another device, the APPLE HDD... (or SSD), with the associated startup volume "Macintosh HD" and others.

I do believe you're going to need to replace that internal drive. It's an involved process best undertaken by a skilled tech. If you're interested in learning about the process, the good folks at OWC/macsales have videos about it.

A good option would be to buy an external ssd drive, format it for Macintosh, install macOS on that and use it as the startup drive going forward.

If you click the View menu in Disk Utility, then click "Show All Devices"... can you see the internal drive now?

If so, you'll need to erase and partition that device before you can install the OS to it.

If not, it may be that the drive has failed and gone away.

As an alternative, you could install macOS on an external drive and boot from that.

Did you replace the original Apple OEM SSD with a third party NVMe SSD? If so, then you won't see the third party SSD because it requires using macOS 10.13+.

Otherwise if you are using the original Apple OEM SSD or Apple hard drive, then @D.I. Johnson is correct and your internal drive has most likely failed and needs to be replaced.