DISTANCES
To measure distances use the Ruler tool.
Find it under the Modify tab.
Or use the Quick Access Toolbar:
Just pick 2 points. The distance will be displayed as a temporary dimension and in the options bar.
You can also add dimensions.
Dimensions only exist in the view they are created in. So if the view you place them is not on a sheet (i.e. a working view) you can leave them there.
Refer to ANNOTATION TOOLS on how to do dimensions.
AREAS
Revit doesn't have a tool to directly measure areas.
Area Schemes and Area Plans are created to measure and schedule areas.
To quickly measure an area draw a Region, select the region and note the area in the Properties window, then delete it.
Refer to DETAILING TOOLS on how to create Regions.
Rooms also measure their areas, so if you want a room area just drop a Room Tag into the room.