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You could probably start at an open-admissions community college in September, if that's what you wanted to do. You would probably need to spend some time there and SUCCESSFULLY COMPLETE YOUR COURSES to build successful reapplication package to a better school.
Any reapplication process would very much involve you proving to an admissions department why and how you became a student who is likely to successfully complete their program and become a graduate that they will be proud of. You need to convince them that your past record is not indicative of what your future record will be. The bad news is that this will not be easy.
The good news is that should you manage to turn things around, graduate admissions committees would be likely to look at you as an interesting candidate who has taken an adverse academic situation and made it better.
Think long and hard about your situation. Even if you treat the anxiety, ADHD, and depression (and I wouldn't recommend trying to start again without dealing with these fundamental problems) there will always be that next problem to overcome. If that next problem will shut you down the same way, you're setting yourself up for the same outcome. Treatment should be a part of your plan to avoid this.
Think about your academic experience, and how that's likely to change. Will future-you be a "B" student, and have an enriching academic experience, or will future-you be constantly on academic probation, constantly on the edge of separation? The former is great, but the latter can be an anxiety-producing monster that will leave you miserable for 4 years, but can still be an important career developing step. Unless you plan on a degree from an open-admissions school (like Brooklyn College, which can give you a fine education that can lead to graduate programs!!), you need to be able to convince a school that you won't fall in to this class, or you just won't get accepted. More importantly, if you think you'll fall into that class anyway, are you up for it? Can you tolerate the long, unpleasant experience an make it through to the other side, and is this worth it to you?
If you plan to continue in school at this point, and you don't plan to enroll in a community college in September, you should be planning on how to keep moving forward and not stagnating. If you need to spend some time getting the ADHD and depression treated, do it. Keep doing things that will develop your maturity and keep moving you to the next step, as opposed to marching in place. If you look like you're not progressing, chances of admissions get even slimmer.
Keeping in mind that an academic experience is a means, and not an end, take a good introspective look at your life plan and career plan and how your education fits in. I highly recommend spending the $4 for http://www.amazon.com/Control-Your-Time-Life-Signet/dp/0451167724/ref=asap_bc?ie=UTF8. It looks like a time-management book, but its much more -- or rather its a time management book on many different time scales -- this moment, weeks, years, decades -- and how to use your time to achieve your goals.