It you are attached to using MediaWiki, the two largest alternatives I know of are Miraheze and ShoutWiki (for some reason ShoutWiki is not accessible right now). I believe both of them allow private wikis. Miraheze is a non-profit so the features and support may be limited. On the flip side, ShoutWiki uses a tiered subscription system. Obviously you get more features and support with the higher-priced tiers. You could also host your own MediaWiki wiki using either a personal server or a generic hosting service. However, I doubt that is what you are looking to do. If you are not attached to MediaWiki then there are lots of other options such as GoDaddy, Square Space, Wix, WordPress, etc.
Edit: Here is a comprehensive (although not necessarily exhaustive) list of MediaWiki farms.