Yes, that's more or less how every Linux "live USB" system works. They all boot from a read-only filesystem image (often a .squashfs file).
The bootloader is not really involved (it gets out of the way as quickly as it can); the OS kernel handles everything.
(It's also how Windows install USBs work – the mini OS really boots from a Boot.wim archive.)
In practice, I guess I am thinking of something like a file-based filesystem. Do such a thing exist?
Any filesystem can be put in a file, as long as the OS knows generally how to attach a filesystem that's stored in a file. For example, you might have seen this done with an .iso file (or a .dmg on macOS). And vice versa, any file format can be called a "file-based filesystem" as long as the OS knows how to present its contents as files. (You could almost say that .zip archives are a file-based filesystem...)
When system boots, this tiny-booting-OS reads the contents of one of those files, loads the kernel into the memory, and the kernel accesses the file, like it was accessing a drive. This file has different segments as directories and files, and stuff.
Yes, many operating systems have a feature that can attach files as if they were virtual disks which behave the same as a real disk as far as filesystems are concerned, so that you can partition them and/or place any regular filesystem in them. Although this isn't always usable for booting the OS, but it's fairly common as far as the ability to store a whole filesystem within a file goes.
Linux allows setting up "loop" devices as virtual disks corresponding to a file. The device can then be partitioned or directly formatted.
OpenBSD and NetBSD have the same function in "vnd" (vnode disk) devices; FreeBSD has "md" devices; Illumos (Solaris) has "lofi" devices; OpenVMS has "LD" devices...
I'm not sure what macOS has, exactly, but it's how .dmg files work – they're not archives; they're compressed filesystem images which macOS attaches as virtual disks.
Windows doesn't seem to have a fully generic feature, but as of Win10 it does allow attaching .iso and .vhd files. Double-click an .iso file (which typically an ISO-9660 filesystem, hence the name) and it'll be attached as a CD-ROM. If you open DiskMgmt, you can create a .vhd file that can be partitioned, formatted with NTFS, and so on.
(Before Windows 10, plenty of third-party tools existed that did the same thing… mainly for CDs though – even back in Win98 days, you'd have software like Daemon Tools or Alcohol 120% to create virtual CD drives for your Legally Acquired Games – but virtual HDDs were certainly a thing as well.)
Of course, not all of these can be used for booting the whole OS – in most cases you need to have the kernel already running and some drivers loaded in order to access the filesystem in which the image is contained.
For Linux specifically, an initramfs is used to solve that problem; it's a tiny filesystem archive that's the first thing to start after the kernel (containing the initial drivers) and contains whatever tools are necessary to mount the real / filesystem. The actual kernel and the initramfs – two files in total – must still live in a location that the bootloader can access (although some bootloaders such as GRUB do support loading them out of a filesystem image as well), but once those are loaded into memory, the initramfs can set up any kind of environment it needs.
Most Linux distributions opt to use an initramfs even for basic disk and filesystem drivers, but it's a fairly small change to make it set up a loop device (or just about anything else) as an additional step. For example:
Nearly all Linux CDs/USBs consist mainly of a file containing a special SquashFS filesystem. (This is actually rather close to being a "file-based filesystem"; it is meant specifically for this kind of read-only usage, with its internal layout designed to pack everything closely together, unlike traditional filesystems that do the opposite.)
You could easily extend this to have several boot menu items, each of which would load a different initramfs and/or specify a different rootfs image as needed.
Some "live CD" builds have a "persistent" mode where a second filesystem image is attached, containing a more traditional writable filesystem such as Ext4, as an overlay to store all changes made to the live environment.
Of course, a regular Ext4 image could be used directly as well. A few years ago, Ubuntu used to have the "Wubi" project which would do a one-click install of the full Ubuntu system into an Ext4 image stored on your Windows NTFS partition.
The bootloader (Grub4dos) had sufficient knowledge of NTFS to find the kernel and the initramfs, while the initramfs then loaded the ntfs-3g driver and set up a 'loop' disk device.
There indeed exist programs that can build you a "multi-Linux" USB stick, at least some of which could boot .iso images dropped into the USB stick's regular filesystem, making the whole "initramfs/loop" process happen twice in a row.
As a completely different example, MS-DOS used to have a "DriveSpace" component which would compress your disk contents; this was also implemented through a driver that stored your FAT filesystem inside a special image file, and during boot would replace the real C:\ with the one stored within the virtual disk.