How on earth is it possible to relax when you are a person that lives with constant anxiety? When every day you are always obsessively thinking about what you aren't doing right, what you should be doing instead of what you are currently doing; and what the next day, week, month, year... etc. will bring?

If you are me, you really don't. I really don't feel like I ever truly relax the way a human being should.
I am always trying, don't get me wrong. Regularly attempting to rest after a long day or shut my brain off during the weekend. Unfortunately, my brain runs about a million miles per hour and the best chance I have had to give my constantly pounding heart a break has been to pop a Benzo.
I know the same things don't work for everyone. I also know it can be very irritating when people suggest so. "Oh, just take some deep breaths. Just meditate. Relaxation exercise. Blah blah blah." Those just don't work for me. I am just TOO anxious. I am thinking TOO much. I have some things that do work though and I am happy to share them with you. I am by no means telling you that these are the things to live by because what works for you, works for YOU but I am so glad that someone I trusted told me about these.
The first one is tapping. I do not know everything about tapping. Please, investigate on your own if this interests you.

How I do this for myself though is I tap one of the points suggested (I like the "karate chop" point with three fingers), I close my eyes to shut out visual shit from distracting me, repeat a mantra to myself like "I may be very anxious right now but I am going to be okay", and to even engage my brain further I mouth it to myself. My best friend it still teaching me about this but this helps me SO much. You can tap on any feeling, just change the mantra.
Another way I relax is since I can't deep breathe just by myself, I find a visual aid. My Fitbit charge has a relax setting that has one. The circle expands and contracts with when I'm supposed to breathe. It also really helps that it vibrates with it too.
Anything to completely distract me really so I can't do that awful thinking thing.
I hope these helped you. If I find anymore techniques that help me, I will definitely share them. If you have any great ones, feel free to share them back!