Yes, it is possible to calculate area in ArcGIS Online. There are two methods to do so:
Using the Measure Tool
The Measure tool uses the Geodesic method, which takes into account the curvature of the earth to create accurate measurements. It does not use any coordinate system information from items in the map to perform calculations.
For more information and detailed steps on how to use the Measure tool in ArcGIS Online, refer to ArcGIS Online: Measure.
Using Arcade expressions to calculate field values
This method uses Arcade to populate an attribute field with area values for different layers. Arcade is an expression language that can be used across the ArcGIS Platform. For more information, refer to ArcGIS Help: Arcade Function Index.
The two Arcade Geometry Functions that can be used to calculate area are:
Note: There are two important factors to be considered before using the Arcade Geometry Functions mentioned above. • The spatial reference information of hosted feature services is derived from the coordinate system of the data frame when the service is published. Therefore, to calculate a field with accurate area measurements in ArcGIS Online for a particular area, ensure the data frame and layers have the desired Projected Coordinate System defined when publishing the service. • Both the Area and AreaGeodetic Arcade functions produce results that are dependent on the extent used to view the feature in ArcGIS Online (output results can only be as precise as the scale resolution).
To calculate area with Arcade: