Database of funded US Department of Defense (DoD) component grant proposals with abstracts: Does it exist?
For most DoD contracts, there is no database or similar system that allows you to get any sort of abstract.  I know this because our research librarians, who track this sort of thing and are quite good at their jobs, have told me definitively that this is the case.

If you really want information about specific projects, you could file a FOIA request, using something like the USA Spending site referenced in Aleksandr Blekh's post.  Note, however, that the intricacies of DoD contracting mean that it may be rather difficult to actually tell which contracts are associated with which programs: for example, many DARPA contracts are actually not concluded by DARPA, but by other DoD organizations acting on their behalf.  FFRDCs and other sorts of special cases further complicate matters.