Currently the default storage backing HDInsight is ASV. You can also store data on the 'local' HDFS filesystem on your HDInsight cluster. However, this means keeping the cluster running permanently, and limits you to the storage on your compute nodes. This can get very expensive.
One solution might be sqoop the results out into something like SQL server (or SQL Azure) depending on size and what you plan to do with them.
Alternatively, I am currently working on a connector between Hive and Azure Tables, which currently allows you to read from Azure Tables into Hive (by way of an external table) but will shortly be getting write support as well.