What is the simplest computational model for which the emptiness problem is undecidable?
Probably you already got these in your bag :-) 


Two way one counter machine over unary alphabet (Minsky61).
Two way weak counter machines (the counter has no effect on the computation but the machine halts if counter reaches zero) [1].
Quantum one counter automata [2].


With binary alphabets, the emptiness remains undecidable for:


One way machines with one unbounded counter and one pushdown store that makes at most one reversal [3].
Two-way machines deterministic finite automata with multiple reversal bounded counters (even over a bounded language) [3].
Stateless (the transitions depend only on the scanned symbol) 2-head 2-way deterministic finite automata even when each head makes only one reversal on the input tape [4].


Edit: on the boundary:


(Open problem) Is the emptiness problem decidable for two-way nondeterministic finite automata with one reversal bounded counter over non-bounded languages? (over bounded languages it is decidable [5])


[1] Tat-hung Chan. On Two-Way Weak Counter Machines. Mathematical Systems Theory 01/1987;
[2] Richard F. Bonner, Rusins Freivalds, and Maksim Kravtsev. 2001. Quantum versus Probabilistic One-Way Finite Automata with Counter. In Proceedings of the 28th Conference on Current Trends in Theory and Practice of Informatics Piestany: Theory and Practice of Informatics (SOFSEM '01), Leszek Pacholski and Peter Ruzicka (Eds.). Springer-Verlag, London, UK, UK, 181-190. 
[3] Oscar H. Ibarra. 1978. Reversal-Bounded Multicounter Machines and Their Decision Problems. J. ACM 25, 1 (January 1978), 116-133.
[4] Oscar H. Ibarra, Juhani Karhumäki, Alexander Okhotin, On stateless multihead automata: Hierarchies and the emptiness problem, Theoretical Computer Science, Volume 411, Issue 3, 6 January 2010, Pages 581-593, ISSN 0304-3975.
[5] Zhe Dang, Oscar H. Ibarra, Zhi-wei Sun. On the emptiness problems for two-way nfa with one reversal-bounded counter. In Proc. Thirteenth Int. Symp. on Algorithms and Computation (2002)