Yes, it does.
Just because it is ASP.NET does not mean that it magically remembers
object data without using hardware. Datasets are heavy objects to
store, as well. Ryu might want to try just storing the tables that are
needed, and not the dataset.
Any time something is stored in session, resources are used. Datasets
have a lot more excess garbage than DataTables do, which makes storing
them in session iffy.
ASP.NET in IIS 5 (not 6) still stores state in a single process. It's
separate from the IIS process, stability is improved, but there still
are issues.