After getting a divorce, will someone without kids have to go back to the barracks?

Each post has its specific rules, but the general rule is that only E6 and above are authorized BAH without dependents. If you are still on a lease you can request an exception to allow you to receive BAH through the rest of the lease. Once you stop receiving BAH you have to move back into the barracks

It's up to your Command. Once your paperwork is finalized & updated BAH with no dependents, then technically you'd not be eligible to keep your BAH, if there's government housing available (i.e. barracks) on the base you're on. As SFC Christopher Smith said you can get a waiver. Since going forward, you'd most likely have to revert to the Barracks & apply for an exception to policy in order to get authorized BAH & live off post.

In my case post regulations allowed (with BN CDR approval) it because I owned real estate locally. Had I been renting I would have been hosed.

Generally speaking, barracks are the standard and any other housing is an exception.