I'd say you're dreaming. The main problem will be the limited RAM.
In 2004, Eric Beiderman managed to get a kernel booting with 2.5MB of RAM, with a lot of functionality removed.
However, that was on x86, and you're talking about ARM. So I tried to build the smallest possible ARM kernel, for the 'versatile' platform (one of the simplest). I turned off all configurable options, including the ones that you're looking for (USB, WiFi, SPI, I2C), to see how small it would get. Now, I'm just referring to the kernel here, and this does not include any userspace components.
The good news: it will fit in your flash. The resulting zImage is 383204 bytes.
The bad news: with 256kB of RAM, it won't be able to boot:
$ size obj/vmlinux
text data bss dec hex filename
734580 51360 14944 800884 c3874 obj/vmlinux
The .text segment is bigger than your available RAM, so the kernel can't decompress, let alone allocate memory to boot, let alone run anything useful.
One workaround would be to use the execute-in-place support (CONFIG_XIP), if your system supports that (ie, it can fetch instructions directly from Flash). However, that means your kernel needs to fit uncompressed in flash, and 734kB > 700kB. Also, the .data and .bss sections total 66kB, leaving abut 190kB for everything else (ie, all dynamically-allocated data structures in the kernel).
That's just the kernel. Without the drivers you need, or any userspace.
So, yes, you're going to need a bit more RAM.
Regarding the ARM Cortex-M3:
Linux requires an MMU (Memory Management Unit). The ARM Cortex-M3 does not have one. It is impossible to run the mainline Linux kernel on the ARM Cortex-M3.
However, there is a variant of the Linux kernel for MMUless processors called uCLinux.
Linux on M3 Guide
ST's Application Note on uCLinux
However, as others have noted, Linux is unlikely to be practical on the M3. It won't run without external RAM. I believe that even the largest Cortex-M3 parts only have 1MB flash, so you'll likely need extra storage too.
Best Answer
STM32's have a serial bootloader built in. See http://www.scienceprog.com/flashing-programs-to-stm32-embedded-bootloader/
STMicro makes a Flash tool to get at the bootloader. If you can find a version that runs under Linux, do some switching to put the STM32 into bootloading mode, and set up uart comms between your chips, you're good to go.
Cross platform Flash Tool for STM32