Final answer:
Barney's status after staying in the apartment without landlord's permission upon lease expiration is that of a tenant at sufferance.
Step-by-step explanation:
When Barney continues to occupy the apartment without the consent of the landlord after the lease has expired and is not renewed, Barney's status is that of a tenant at sufferance. A tenant at sufferance is someone who remains in possession of a property without the landlord's permission after the expiration of a lawful tenancy. This situation often arises at the end of a lease term when a tenant fails to leave the property, and the landlord has not taken action to evict the tenant or to extend or renew the lease. As a tenant at sufferance, Barney has no legal right to remain in the apartment, and the landlord could begin eviction proceedings.