I don't have the codebase for Ask Ubuntu so I can't really say if there's anything particularly platform specific that would cause a blocker but in answer to can you migrate it, the answer is probably yes.
You have to keep in mind that the platform that ASP.NET websites sit on are more than a magical ASP.NET binary:
ASP.NET can largely be replaced by Mono. It's not always pretty and the performance isn't always great but it can be done.
ASP.NET MVC is the templating engine. It normally sits on top of ASP.NET but it can run on Mono with a few changes.
IIS could be removed fairly simply, allowing for custom handlers that SO have running on it, replacing it with Apache (et al).
The database is technically the biggest part of the whole. In simple cases, where you're using good code, sure. Rip out the database and replace it. Specifically for here, the cluster arrangement of MSSQL servers that runs the StackExchange sites could probably be ported to MySQL (or other) but it would probably need significant work as there's probably a fair portion of DB-specific code lurking around, either in the database itself or (less likely) in the site itself. Most code-based, database-specific data-faffing is a thing of the past thanks to LINQ, which Mono does support (for non-SQL Server databases too).
That's by no means a definitive list of what would be replaced but they're the major items.
Whether or not you'd want to... That's a tougher question. By the time you'd hunted through a few dozen issues, you're probably already at the point where you could have just reimplemented the system in something like Django. Of course that depends on the complexity of the system. It's as hard as a piece of string.