The Minnesota state government?
Well, that, too (as of midnight last night).
But the shut-down I was referring to is the Edina Realty City Lakes office, effectively closed by an out-of-commission A/C unit.
In fact, the building’s a/c has been fitful for the most of the almost-decade I’ve been in the office.
Just one more reason that Realtors in the office are counting down to the opening of the new office a couple blocks to the west, on Minnetonka Blvd.
The Other Shut-Down
As far as that other shut-down, the effects on state real estate transactions are likely to be muted, at least in the short run.
That’s because local, not state government, plays a much bigger role in real estate transactions.
That includes county recording, district courts, municipal truth-in-sale inspections, etc.
However, if a closing requires the release of a state tax lien on a property, it may not be possible to obtain the payoff amount or acquire a tax-lien release during the shutdown. 
Of course, if the Buyer in a pending sale is a state employee, their ability to come up with the funds while effectively on unpaid furlough is questionable.
If the shut-down drags on, look for the parties in such transactions to negotiate extensions on any applicable deadlines (closing, written statements, etc.).