What people remember -or have heard- about the Vietnam draft was that millions of young men were drafted to fight the war, millions tried to get out of it, sometime fraudulently, and millions fled to Canada to avoid it. The real story is not quite so extreme. Only about 25% of the eight million Americans who served in Vietnam during the war were draftees, and only about 40,000 young men fled to Canada to avoid the draft.
Not every eligible young man was in danger of being drafted, since the Selective Service had a lottery to determine who would be called up first- by the date of their birthdays. These lotteries were televised, so young men could witness their odds of being shipped overseas in real time. While the lotteries were fairly random, getting a student deferment or a medical exemption was easier for those with money and connections. The threat of the draft hanging over one's head for years led to massive cultural shifts in the 1960s and '70s. Read the facts behind the Vietnam War draft at Mental Floss.
In psychiatrist Major Sidney Freedman's first appearance in Season 2 Episode 3 "Radar's Report" he diagnoses Klinger as "a transvestite and a homosexual" and Klinger flatly refuses to sign the report, which would have given him the discharge from the Army he wanted.