There's one reason I'm posting this here: it made a difference for me and I want to make a case for this so that maybe it proves useful to some other people.
There are two reasons why I think running is good, but first I want to talk about all the stuff that's keeping you likely keeping you from running.
I'm not a runner, never have been. I was skinny fat when I started, but I think this gives you the same handicap if you're fat: you have no muscle strength, no endurance, your tendons and ligaments are weak. When you start running, it feels like fucking dying. Everything from your legs to your abdomen, chest, and even your head will hurt. But here's the thing: every single time you do it, your body, all automatically, changes it self a little bit to avoid this pain. Every time you subject yourself to this torture, your body does everything it can to make the next time easier. It's biological, it's guaranteed. As long as you do this, even once a week, this happens.
With that out of the way, here are the reasons why it's good:
It does something to your psyche. I don't know whether it works the same for everyone, but it's a big enough effect that I'm writing about it. It sort of reminds me of meditation - it clears your thoughts. I started doing it because I was depressed as fuck and it helped out a lot. Coupled with calisthenics, I think this is what pulled me out of that dark pit and set me doing things I always wanted to do in life.
It's good for your brain - https://en.wikipedia.org/wiki/Neurobiological_effects_of_physical_exercise
. In the short term it helps your brain cope with stress. In the long term, it improves your cognitive skills like memory and focus. It does this by changing your brain chemistry so that your neurons get more resources. Finally, it improves your cardiovascular health - this is mentioned so often that I think it has lost its meaning so I will describe it like this: you'll be able to go further in any physical activity and you'll live longer. Let that sink in - you'll be able to, completely on your own, transport yourself further or perform physical activities for longer. This is literally like leveling up your character stats.
Ok, let's get down to actually doing it.
Running is cheap to do. You can start with what you have now (any shoes, cloths, etc.) and once you scrounge up enough cash you can buy yourself a pair of decent shoes (40-100$). Decent shoes are ones with little to no cushioning ie. you're looking for the flatest, most flexible shoes in the running section. This will help you avoid heel striking and a lot of pain. Light shoes make you naturally avoid heel striking - you'll feel like you're landing on the balls of your feet instead. Your stride will get shorter but you will still run at the same speed.
How do you start running? I recommend using couch-to-5-k because it gets a lot of things right. Don't worry about making it to running 5ks in 8 weeks - treat the program more as a guideline. If you can't run, start with the walking week. If you can run a little, start with week 2 or 3. Don't overtrain - stop running if you're feeling physical pain. You want to give your body time to adjust. This is especially important if you don't do any physical exercise: your soft tissues will adjust fairly quickly, but your ligaments and tendons need weeks/months.
If you wanna do this, just get up, put on your shoes, and do it. You might want to do a small warmup first - do a few squats, run in place for 15-20 seconds, anything to get your muscles warm. Are you afraid of other people seeing you? I've been there. I solved it by running early morning/late at night and after a few runs I decided to run in the day. You know what happened? Nothing at all. People don't care about other people. They barely spare you a 500 millisecond look.
Like I mentioned earlier, your whole body will probably hurt at this point. But you did it and be proud of every little victory like this and prepare yourself for the next run. Try to keep runs a day or two apart to let your body heal. If you're starting from zero, getting to a 30 minute run might take you anywhere from 3 to 6 months, but that's just a goal - all the good stuff, like getting disciplined, building a stronger body, relieving stress happens all the times - both when you're running and when you're healing.