Not read all the replies but you absokutely can run if you want to.
I hadn't run since being forced to at school. I hated running and was unfit but randomly decided I wanted to do the Great North Run. Within 18m of so/so committment I had run the GNR and a marathon and loved running (most of the time!).
Then I broke my arm so had to stop for a while and have other stuff I've had to deal with so haven't restarted.
When I started running, I was exactly the same as you. I started with the c25k but tweaked it so I followed the principles but not the actual time table iyswim.
The first few weeks were so hard every single time! It was elated when I ran a few meters and didn't die!
Then it just suddenly clicked.
Even when I was running 10+ miles twice a week, the first 10mins or so would sometimes be a bit hard. It was like I just had to wait for my body to say "Ah! We're running. Let's get into running mode". I had to force myself through that and then my body would settle and away I'd go. It's the same when you're starting.
There are also the runs where you just can't get into a rhythm. For whatever reason you can't get your pace right or your breathing is off or just something. Those are hard runs. When you start running you are teaching your body how to respond to a new activity so for a while, every run will be a hard run.
Also, look at your pacing. I am not a fast runner. If I try to go fast I kill myself. It took me a while to find my ideal pace and again, until I did that, it was hard.
Set a goal, something like a 5k park run in a realistic time frame. Don't think further than that. WHEN you do it, you'll feel amazing!
Be warned though - I was at my heaviest when I was running. I wasn't eating any differently but my legs bulk up very, very easily. I wanted lean runner legs, I actually got thick strong rugby player legs! I was really not very happy about that! Haha!