If so, to what extent and why?
I believe that women are still restricted from significant ground combat roles.
Are there other examples or is that not really discrimination?
If a private company tried to use a test like theASVAB to sort out applicants, they would probably get sued for racial discrimination by disparate impact. African American and Hispanics on average do quite a bit worse than Whites and Asians on these tests.
Congress writes Federal laws and may draw such lines as it sees fit, consistent with its constitutional power to “raise and support Armies… to provide and maintain a Navy; to make Rules for the Government and Regulation of the land and naval Forces;… [and] to provide for organizing, arming, and disciplining, the Militia…” Art. I, Sec. 8, cl. 12-16. Women are still, by law, kept out of some combat roles, but of course the battlefields of Iraq and Afghanistan have blurred the distinction between the front lines and the rear echelons.
Looks like DADT will be repealed with the President’s signature today, but gays were obviously discriminated against by Act of Congress while it was in effect. Although the Federal courts are the ultimate guarantors of individual rights against legislative abuse, they have historically been very deferential to Congress’s discretion in passing laws affecting the military, and the President’s discretion in interpreting and applying those laws as commander-in-chief.
Not here (Canada). Women serve in combat roles and I personally worked with a Sgt that underwent gender-reassignment surgery while still working.
There are a lot of things beyond your control that can bar you from the military that wouldn’t be legal if it were a company, such as having flat feet or a history of migraines. To a lesser degree, a history of drug abuse or mental health treatment. Incidentally, this is why I rank the importance of DADT waaaay below the importance of gay marriage- they keep people out of the military for all kinds of silly “just might” reasons.
But anyway, yeah, the military’s special.
Having flat feet or a history of migrains is not a protected class. A private company (in the US) can legally not hire you for the same reasons. The same goes for history of drug abuse or even mental health treatment. A company can legally use that as a reason to not hire you.
There are are only certain criteria that a company may NOT legally discriminate against when hiring:
Both flat feet and migraines can fall under the provisions of the Americans with Disabilities Act.
So can obesity, but it’s still used as a disqualifier for military service in the US.
Right. I am disagreeing with Wilbo.
And sexual orientation, despite frequent proposals otherwise, is still not a protected class under Federal nondiscrimination law.
I know, I’m supporting you with another example. 
Do you mean to imply that obesity shouldn’t be a disqualifier
Is it true that the military (or certain branches at least) won’t take you if you fall outside a fixed height range?
A private company can refuse to hire people with flat feet or migraines or who are obese, if (and only if) they can demonstrate that those disabilities prevent the candidate from performing essential functions of the job with or without accommodation.
Presumably, if I were hiring people who were required to march 20 miles a day with a 100-lb pack, I would be justified in not hiring people with flat feet and would not roun afoul of the ADA.
I suffer from migraines and am currently in full-time service. The migraines alone will not get me booted - but if I fell outside of ‘universality of service’ because of the migraines, then I could (and would be) released.
BTW - universality of service means meeting the physical requirements if being a soldier.
It can really suck to be carrying a ruck sack while having a migraine, but it won’t kill me, just makes me sick.
cite?
The Marines certainly do, and I’m sure the Navy and the lesser branches do. 
They have to buy those uniforms in bulk, I’m sure.
Cite that obesity is a disability as defined by the law?
Not being a lawyer, I provide a citation rather than an answer.
Not being an Elf, I will say that I am dubious of this.
When I was in the service, they were at pains to tell us that we had not lost any of our civil rights, but in fact, who determines what constitutes ‘reasonable’ search and seizure, or ‘due process’ had changed.