Yes, you can. You can download your database at any time by going to Admin > Your On Demand Account and selecting a download format. After that, you can point a local install of FogBugz at this database and it'll run right away. For detailed instructions on this process, please see Importing data from FogBugz On Demand or trial into SQL Server.
Note that Kiln On Demand can not currently be easily moved from our servers to other servers. You can however, get recent clones of your data and push them to your Kiln server.