For requests like these I would probably push back and try to get the request scoped out further. What is the underlying business need? What insights are the end users hoping to get out of the view / dashboard? If they know what they are looking for beforehand, would there be a set of views (each with some lesser amount of flexibility) that would let them get at those insights more quickly?
If the end users don't know what they are looking for and really need more of a sandbox, I suspect the preferred way to handle this request would be to provide them with editor (explorer?) licenses for Tableau Server and let them edit a viz themselves. You could give them access to a curated data set with only the dimensions and measures the end users would need, then let them play around with the different fields and how they would want the data displayed.
With all of that said, you can probably build a view with 5 "variable" dimensions (calculated fields that each use a different parameter to determine which of the 5 dimensions' values to use), then create cascading filters for those fields, but it will be messy.
EDIT: added attachment to show how variable dimensions could be implemented
SampleVariableDimensions.twb 236.6 KB