[Verginius Rufus]

Even if Congress passes such a law, Obama will veto it, and enough Democrats will support his veto that it will stand.

What is the legal status of the base? Is it something that the Commander-in-Chief can decide on unilaterally?

[AFreeBird]

No, congress has a say. Although as CIC, he could theoretically order all military people out and abandon the place. But congress could intervene.