I need to get the URL attached to a file in a saved search, is that possible?

use a join for File (line) Fields or File Fields > URL (depending on file attached in line level or header)

can you do this on any type of search?

as long the record can have attachment - this is available

is this in the criteria of a saved search?

nope in the result tab, when you select the field for the column scroll down in the drop down list for the field selector till "..." anything below are join option for other record types. Look for the file field and select it another pop box will open and will let you select the desired field/fields from that record type