The FAR, DFARS and AFFARS do not go beyond Ratification of unauthorized commitments provided for in FAR 1.602-3 and the supplements.  Both the Government Services Administration and the Defense Logistics Agency include requirements for quantum meruit in their supplements.  (GSAM 501.602-3 (f) Payment based on quantum meruit or quantum valebant and DLAD 1.602-3(S-91) and (S-92) - Ratification of unauthorized commitments.)

Numerous GAO rulings have indicated that providers to DoD entities may be paid under a quantum meruit basis.  Several criteria have to be met before this can happen. 

1. The government must have received benefit from the services.  In this case the government recognizes that it has received and benefited from the services provided.
2. The services provided could be obtained through a permissible procurement.  In this instance, there appears to have been a permissible contract in place, and there was some confusion as to when and how it was to be terminated.
3. The cost matches the fair and reasonable value of the services. The Contracting Officer will have to make this determination.
4. The parties acted in good faith.
5. Funding is available

See:
GAO - B-209173 - Payment on Quantum Meruit Basis in Absence of Formal Contract - Jan 17, 1983 
GAO - B-200372 - Authorization for Payment on Quantum Meruit Basis - Nov 18, 1980
GAO - B-242405 - Invalid Intergovernmental Personnel Act Assignment - Quantum Meruit - Mar 15, 1991 

Another possibility to resolve the situation could be based in further reading of FAR 1.602-3 Ratification of unauthorized commitments:

FAR 1.602-3(d) Nonratifiable commitments. Cases that are not ratifiable under this subsection may be subject to resolution as recommended by the Government Accountability Office under its claim procedure (GAO Policy and Procedures Manual for Guidance of Federal Agencies, Title 4, Chapter 2), or as authorized by FAR subpart  50.1. Legal advice should be obtained in these cases.

This refers us to FAR Subpart 50.1 - Extraordinary Contractual Actions and the safety act – FAR 50.103-2(c), with the associated parts of the DFARS PGI 250.103 and AFFARS 5350.103, which may provide another avenue to allow justification of payment without a contract being in place.