To confuse things a bit more, even if OSHA gained access, and noted violations they could cite the base, as the employer...however no fines could be imposed. More correctly, fines could be levied, but they would not have to be paid. The military, being a government body, cannot be made to pay a fine by another government body.
DOT is a prime example. Iowa is a state program, so the only inspections performed are state OSHA inspections. If they note a problem at a DOT worksite, they can gripe all they want, but can't fine DOT. Something about taking money from one pocket and putting in another...kinda makes sense. Buddy's a heavy equip operator for Iowa DOT and Iowa OSHA once tried to give a NOV on their site based on shoring of excavations. Didn't fly, but at least the violations were fixed. (Buddy actually called OSHA in.)
Now, were a fed OSHA inspector get peeved at a state DOT site, who knows?