How can the blockchain survive without an army or other kind of law enforcement?
The blockchain itself does not have any power. It is just public set of records, like a phonebook. 

Depending on the contract between trading parties, you arbitrage the contract disputes within the legal means you have available. 

However, blockchain can act as a proof of agreement and everything signed there is very hard to forge: the other party cannot claim they did not sign the agreement if the private key used in signing can be proven to belong to the party.