You can't run DOS on an ARM machine. DOS is inherently tied to the PC (x86 w/ BIOS and specific well-known IO/memory locations) architecture. You can, however, run Linux on larger ARM processors quite readily, and there are usually ready-made ports for these development type boards, and MAME runs fairly decently on Linux.
You got something that backs that up?
Um, that's just how DOS works. The wikipedia article explains this to some degree (read the "history" section). There may be "DOS like" OSes for ARM systems in that they have a similar shell and maybe a similar API, but it ain't actually DOS and can't run any DOS binary you'll find. Even the "DOS prompt" in WinNT (NT4, 2k, XP, Vista, 7) isn't really DOS running as a typical OS. It's either a virtual machine (read: emulator) running DOS, albeit with minimal virtualization (command.com in 32-bit NT, non-existent in 64-bit NT) or a native Windows application that's just a shell designed to look and fell like DOS (cmd.exe in all versions of NT).
As to why you can't run DOS directly on an ARM system:
First off, DOS programs as well as DOS itself and included utilities are all x86 binaries. They will not run on ARM without an emulator. The instruction set (and machine encoding) is completely incompatible.
Second, DOS itself makes HEAVY use of the PC BIOS, and the PC BIOS doesn't exist on anything other than a PC. There in fact exist x86 systems that are not PC compatible and don't have anything like the BIOS (instead just having a lightweight bootloader that provides no facilities other than the ability to load a "real" OS). DOS was in fact basically a thin wrapper around the BIOS with some libraries for e.g. filesystem access and memory management thrown in. It didn't really do that much. While there was some intent to make DOS run on x86 devices that didn't have a BIOS, this never actually happened as there was little to no demand for it.
Third, many DOS programs pretty much bypass DOS itself and bang directly on the PC hardware using the "well known IO and memory locations" inherent to the PC architecture. Essentially, they include their own drivers that assume an underlying IBM PC compatible system. None of these will be the same on an ARM system and will in fact vary depending on which ARM SoC is being used.
You could patch all this up by running a PC emulator, and in fact such emulators exist. Bochs will run on ARM, for example. However, emulation of an entire PC, including the CPU, is not without substantial performance cost. It also seems silly to emulate a PC just to turn around and emulate something else using MAME, in this case, especially given that one could just build a native version of MAME on e.g. Linux. You'd need an OS in which to run the emulator, anyway.
One could, of course, port most of one of the open source versions of DOS to ARM if one REALLY wanted to. It would be quite an effort, and it still wouldn't run any DOS programs in existence (due to the instruction set incompatibility). Again, it would be more of a "DOS-like" OS than actually being DOS. Since there exist more modern OSes that are already ported to many ARM systems and provide much more functionality, including ports of many popular applications such as MAME, with minimal overhead, there's little reason to do so.