Mapping Center Answer:

You will need a feature class that defines the area (e.g., your study area or map sheet extent) that you'll want to base your area calculations on.  

Use the Feature to Polygon tool to produce the polygons which will automatically have an area field.  If you want the area in a specific unit (other than your coordinate system's units), add a new double/floating point field and then Calculate Geometry.  (You can also automate this in the Calculate Field tool by using a Python expression.)

You will still need to manually select the polygon you want to know the area for, based on the contour elevation values.  

If you want to automate the selection for one particular elevation range, do the following.

Select by Attributes the contour elevation for the minimum value in the range, e.g., "CONTOUR" = 1400

Then Select by Location the features in the result of the Feature Polygon tool based on intersecting the selected contours.  This will select the polygons on both sides, so the next step is to to unselect the polygon on the wrong side of your range.

Let's say I was interested in a range of 1390 - 1400.   But right now I also have polygons selected that intersect 1410.  So, Select by Attributes on the Contours for "CONTOUR" = 1410.  Now use the Select by Location to "remove from the currently selected features in" your polygons.  

That will give you all the polygons in the range you are interested in.  If you open the Attribute table for your polygons, you right-click on the area column and choose Statistics and get the sum (among other summary statistics) for the area of that range.

The point in sharing that is that you could build a model to automate that where inputs are your contours, a boundary, and a range (min value and max value).