Is it possible to set up a custom report with multiple tabs, and have each tab filter data differently? I have too many metrics to use a dashboard.
No to tabs, but you could use a combination of multiple dashboards and shortcuts. Shortcuts will let you save certain views of reports (such as secondary filtering data) allowing you to access reports that would typically be nested a few clicks deep. You could for example filter on a specific keyword by referral path. Shortcuts could also be used on any custom dashboard to allow for quicker access to the data you are looking for.
Shortcuts would also be available on the top left hand side of the page, which may speed up the process of finding the data you're looking for.
No, the filters in a custom report do apply to all metric groups in the report (which is propably what you mean by "tabs") - the different metric groups in the report are supposed to apply to the same subset of data, which is why the filters are the common element for all tabs.
Edit: Sorry, you mean of course report tabs (that the other anwsers insist do no exist, even though the option is called "report tab" right there in the interface). However my answer still applies.