Archimedes were home computers competing with 16bit Amigas and the like, existing in the transitional period when Wintel boxes that did useful things were becoming affordable. A very comprehensive photo record can be found at Chris' Acorns for those wondering which home computers I'm talking about.
They were pretty great for their era: a simple GUI, very easy for even a kid to program quite complex apps with, font anti-aliasing that was well ahead of its time, an intuitive vector graphics package called !Draw supplied with the system, and everything making good use of the limited resources available to computers.
We moved directly from a Spectrum +2 to an A3000, upgraded that to 4Mb, and from there to an A3020. Clinging on to that lasted until I needed something for college and uni and — after considering a RiscPC for a short while — I opted for a cheaper and more functional 233Mhz K6-2 era PC running Windows 98. At the time I was just getting interested in web design, and Acorns lacked a virtual memory backend for modern applications — if you think about it, the average webpage uses a frightening amount of memory taking into account the need to uncompress and display images. Besides which, using RISC OS even back then (1999?) involved running to stand still rather than to advance, much like an old PC gradually consumes more time trying to run the software typical users find themselves running day-to-day.
By now a feature advocacy list for RISC OS reads like an exercise in expressing desperation, though some commercial RISC OS software is still being developed and a few apologist stragglers hang on. To most intents and purposes, though, the platform is either dead or an evolutionary brick wall — a ton of freeware and open source software is available for other platforms, and if you're really attached to the RISC OS desktop you could customise a Linux distro to act like it. Reusable skills are key, after all.
Emulation is a trickier legal area than for some shelved platforms — whilst some software's been explicitly freed by authors considering it no longer a commercial going concern, many coders have moved on and wouldn't be contactable by the few individuals maintaining online archives. It'll probably be a good few years yet (as it took for Spectrum emulation) before people take the risk of just putting everything up and inviting those who still care to issue takedowns. And before those who're inclined to issue takedowns, more out of pique than any commercial interest, will quit bothering.
There have been and are commercial emulators with approved ROM images included. VirtualA5000, which would be the most useful for hobby users, is no longer commercially available but turns up occasionally on eBay and patches to the last released version can be found here. The current VirtualAcorn product, VirtualRPC, is more expensive than a real RiscPC would be, and both are incompatible with a fair amount of older RISC OS software (notably games, which is what emulation fans tend to be most interested in.)
Since it's no longer for sale, it's a shame that VA5000 hasn't been released in the same way its predecessor RedSquirrel was. That emulator project has been abandoned, like Archie and several others. One still kicking is Arculator, recently joined by RPCemu.
As with most emulators, you'll need a virtual image of the machine's stored-on-chip core code. You can copy ROMs from a working machine easily enough (or if determined to make a point, mime cutting up one of the dozens you junked back whilst doing school network admin stuff and toss the chip into a drawer, then just go and download the contents...) Note that you'll need a three-button mouse to work with RISC OS; many wheelmice work fine, but aren't comfortable for much GUI usage. For anyone hoping to indulge in nostalgia other than games, an old/cheap three-button serial mouse might be a good idea.
Following visits to the Arculator homepage and RetroBBC if you hunt around a bit old classics such as 2067BC and Mad Prof Moriati aren't too far away... or, if you're like me, rather more obscure magazine cover-disc fare such as Car Wars or Moonquake.
Also of interest: it's possible an open-source RISC OS project will bear fruit (or, indeed, acorns) in future for those looking to emulate, though again there are likely to be compatibility problems with old software. Could be worth a bit of tinkering if/when it reaches the pre-rolled binaries stage.
Those of a less technical bent might like to try the now-freeware Spheres of Chaos.