Ariadne has an inherently stable design. Not in the least because of PHP's shared-nothing design, which means that almost any problem a single user encounters will not affect any other users. As long as your server is running, Ariadne is running also. There are no seperate processes. We've been working on Ariadne and running Ariadne based websites since 1999, with a continuous upgrade path, without any major problems.
Ariadne has been designed primarily for Apache and MySQL, which are very stable and relatively easy to administer. In addition Ariadne incorporates some checks to prevent you from making a number of errors. Any template you save is first syntax checked. If that fails, the template is not compiled and Ariadne runs on using the old template while you can fix your mistake.
For guaranteed stability you'll need to invest in development and staging servers, which Ariadne supports through SVN (the Subversion version control system) and its import / export mechanism. Its also fairly easy to configure a single Ariadne system to run on multiple servers.