I've found myself in this situation more than once, so I figured I'd ask to see if there's a way.
If I have a machine (with an OS), can I somehow mount an ISO somewhere on the hard disk, have the boot manager (grub/windows boot manager) pick up that ISO, boot off of it, and install the OS onto that machine? All while keeping any important data on the drive, but assuming there's enough space.
What if the machine has two hard drives? (One is empty in the case that caused this question).
This is possible. You can install different OSes on different drives. You can also use hardrive partitioning to create logically seperated drives on the same physical drive. To do such an installation you would need to do your partioning up front and then start installing your OS. Most installers allow you to select preferences such as on which drive you want to install.
Another option is to use virtual machines. Like this an application running on your machine and on your OS provides a virtual machine exposing access to hardware or emulating hardware. Like this you can effectlively run multiple OSes at the same time all as applications in your host OS.
It depends on how you want to use them but it sounds like what I have would work for you. I have multiple HDDs with a panel mounted power switch so I can turn them off when not being used. I have a backup OS on one and a W7 on another. I also use them for "offline" data backup. I just turn on the one with the OS I want to boot from or the one I want to backup to. The ports are set for hot swap so I can turn one on while operating to backup to.
There's a few approaches to do this and it depends on the OS you're starting with. There's a bunch of posts about bootstrapping windows from windows - this worked well for me, and it still mostly works on windows 10 IIRC. While it focuses on a VHD install, there's no reason it wouldn't work on a real partition. You can bootstrap an installer for linux from windows (using easybcd - which uses grub4dos for this).
Likewise you can bootstrap a linux iso from grub for an install, though I'm not aware of any way of doing something similar for windows.
In the context of modern systems, in most cases, outside systems you need to routinely wipe and reinstall from scratch, a disk image or USB key set up for unattended install is likely going to be simpler.
