Can a building that's been owned in a business for 34 years still claim a depreciation deduction, or has that run out?

Is it still being used by the same business or is a new business using it?

UPDATED FOR TAX YEAR 2019

Probably.

Depending on the year it was put in service in the business, and the system of depreciation used (ACRS or MACRS) it has a depreciable life of 15-35 years 

If you placed a non-residential building in service in 1985, it would have a recovery period of 18 years under ACRS and would be fully depreciated.

[Edited | 3/26/2020 |  11:51am PDT]