Unfortunately, no. Burials are now closed at most of the National Cemeteries maintained by the park service/Dept of Interior. (See here - there's two left open http://www.cem.va.gov/cems/doi.asp ). Arlington is maintained by the Army. There are many other national cemeteries available, some are closed, others are still open. The website I pointed you to before has more details. The criteria used to be a bit stricter at Arlington than elsewhere, but I think it's been relaxed a bit. The issue primarily is "active-duty". They've created a large columbarium which gives them quite a bit more "space" to inurn/intern remains for those whe served and were discharged honorably.

The unnumbered/nameless you see were from the Battle of Gettysburg only.

What can and may be provided at any burial depends upon the rank of the deceased. If you're a veteran, check out the veteran affairs website so you can be aware of what's available to you.