Odds of Getting In
Decent. But it depends on what you mean by "union," and it depends a lot on the political climate of your era. Obama's presidency will be marked by unprecedented capitulation to unions, from the General Motors bail-out to dozens of new laws that make the business climate much more union-friendly.
Have a law degree? It'll help. Speak Chinese? It won't help at all. Good at swinging a Louisville Slugger? Not relevant today—that's Unions 101, Old School.