Greetings,
The runuo server will run on most computers that are capable of running the current .NET framework (which is freely available from microsoft).
There are however a few things to keep in mind when dealing with the runuo server which are (in no particular order):
a runuo server requires a substantial amount of ram (512 is the current recommended minimum). After that any half-way decent PC (state of the art is not necessary) should be able to run the server.
a runuo server requires a decent internet connection in terms of _upload_ speed. Every player on your shard eats anything from 2-5kb/sec in upload speed (once more, this isn't download speed).
A runuo server eats bandwidth for breakfast. Depending on the amount of players on your shard the traffic to and from your server can go through the roof. (This is very much dependant on what you do with your server though).
Runuo needs the .Net framework... this may limit the type of server that can be used for running this wonderful UO emulator.
Home hosting and professional hosting are both good options to run your server on provided you have a decent internet connection are both capable of running a server without problems.
Whether your own server is sufficient to run the server depends on the things mentioned above, I am sure you can make a decent judgement call on that yourself.
Hope that helps.
*edited*
Forum member Phantom recently awarded me with a substantial amount of negative reputation over this post. Since his comment was that I do not know anything about the runuo server and shouldn't give advise I decided to pass on this information.
I have no intentions of providing incorrect information. Readers of this post please be aware that I am not trying to purposely mislead anyone. You may contact forum member Phantom to verify his opinion about this post and to retrieve what he believes to be correct information.