Can't see my hard drive only disk0 and OS X Base System listed on Disc Utilities.

To find the hard drive, you'll need to make sure it has been properly reformatted.

Once you're booted to recovery mode and facing the MacOS Utilities window, select Dick Utility and launch it.

In Disk Utility, make sure to click View > Show All Devices. You should now find the internal drive device listed in the sidebar, usually the top most device and named Apple SSD....Media. Select that device and erase and format it with APFS format, GUID partition scheme and name it "Macintosh HD" (or SSD... whatever you like).

Then run the First Aid on the drive. If ** gives it the green light, quit **.

Please find additional guidance here > Erase and reformat a storage device in Disk Utility on Mac - Apple Support

Now you can select Reinstall macOS from the macOS Utilities screen. It will guide you through the steps to select the drive you just reformatted and reinstall the OS.

Please see the guidance here > How to reinstall macOS - Apple Support

Or your using a third internal drive. If so, then you need to be booting from a macOS 10.13+ installer. Your picture of Disk Utility indicates you are using an older version of macOS 10.6-10.12 which is not able to recognize a third party internal NVMe SSD.

FYI, it always helps to provide the exact model of your Mac and previously installed version of macOS, and also any hardware modifications such as a third party SSD. Plus it helps to know why you are needing to reinstall macOS in the first place since the two issues may have the same source. You can get the exact model of your Mac by entering the system serial number here:

Check Your Service and Support Coverage - Apple Support

Also, it would be nice if you actually rotated your pictures into the proper orientation before posting them. It is so difficult & annoying to view sideways pictures when assisting people....many times I will ignore the sideways pictures. If you are using an iPhone or iPad, then you can rotate the pictures by using the instructions in this article:

https://9to5mac.com/2019/01/09/rotate-photo-photos-app-iphone-ipad/

I'd get an external drive to Install to, looks to me like the internal drive is gone‽