I don't chime in on this site often anymore, but I will give you my input. An ELS discharge, like said before, pretty much means you were not in the military legally. You did not serve for 180 days on active duty so you are not considered a veteran. Where you screwed yourself, is being booted for disciplinary reasons. Most government agencies INCLUDING police departments, will look at people with ELS discharges on a case-by-case basis, UNLESS it is for disciplinary reasons. So no the ELS should not hurt you, but the reason BEHIND the ELS will kill you. What is your SEP code? It will say on your long form DD214 or NG22 (I think that's what the guard calls it).
Will this stop you from getting regular jobs outside of EMS? Nah. Will it stop you from getting most transfer service "EMS" jobs? Probably not. Will it stop you getting a job with a reputable department that you can make a career out of? Probably.
Also as others have said. You must have been screwing up more during basic and pissed people off, because when I was in OSUT as an 11x, people fought ALL the time. Punishment? Maybe an article 15 and their wrists would be tied together with 2 feet of 550 cord and told to deal with one another for the day. Stop fooling yourself by saying you're SDS said you would make a good infantryman. Because guess what? They are the ones who referred you to be kicked out and did not stand up for you when it went up the chain. They could have even recycled you back to day one, or a certain point if they thought you just needed some time to be whipped into shape. Instead they sent you home. Did you complete all the training up until you were discharged? Or were you a sick-call-ranger as well? 3 days before graduating and you best believe they would have kept you if you were not FUBAR. So please do yourself a favor and stop falling into the ELS norm and trying to say you would have been a good soldier, and that others didn't deserve to be there. Guess what? YOU got kicked, and they did not. Hence they thought otherwise of your performance and character. Every ELS guy says this, and its annoying.
If you take what I said and get all bent out-of-shape about it...then you need not apply to public service. Do what you are doing now and put distance between it. Get a 4 year degree, and then try.