If you mean "no external filesystem on a block device" support like ext2/3/4, vfat, etc. you can build a kernel without any filesystem modules either built as modules or built into the kernel. I'm not sure if the
initramfs support counts as a filesystem - you'll need one of course if you want a shell and other utilities. You can use
tmpfs for things (and of course, this is really a RAM filesystem) like
/dev. So you can run Linux with only the
initramfs mounted on /, as well as things like
/proc and such.
Any Linux software will at least minimally reference the filesystem in some way, it's a central concept of UNIX and thus Linux, so you can't get rid of the whole notion of files in Linux and it still be Linux.
You used to be able to disable the block layer when compiling a kernel. I don't know if it's still possible.
The MMU is a CPU hardware feature and Linux requires it. You generally want this as it enables processes to be separate from each other and most of the security aspects of the kernel.
Nonetheless, a project called uCLinux exists which is an implementation of Linux not requiring an MMU. It's meant to enable a Linux-like environment in microcontrollers and such. Processes in uCLinux can overwrite the address space of others, leading to a less stable system if you don't have well-behaved software.