For a period of about 6 years, ending a couple of years back, I was juggling a full-time job with eldercare responsibilities for my mum and both of my in-laws, all of whom had mental health/dementia issues to one degree or another. I was a 20 stone, middle-aged man with a lot of stress at work and next to no down time at home. I honestly think if I hadn't taken up cycling I would have had a heart attack or a breakdown years ago.
Sometimes I had to settle for just cycling around the block a few times, never being more than 10 minutes from home in case of an emergency call out, but the very fact of being on my own on the bike, away from all the stress for half an hour's "me time" was wonderful. As and when other relatives took their turn at caring duties, I started slipping away for days or half days with the car and going for rides with friends in more and more beautiful and interesting places. Cheddar Gorge, the Strawberry Line, the New Forest, the Isle of Wight, the Forest of Dean , Bristol to Bath Railway Path, the Cotswolds, etc .... and I could feel all the stress lifting from me.
My wife and I later both went part time, and with the passing of all our parents, life has settled down to a much more stress-free level. I now cycle not because I have to, but because I love it and I recognise just how good it is for me. Not only does it get you fit so that you can cope with stress better, but it actually reduces stress by taking your mind off things and/or giving you time to think things through on your own. For me, cycling time is quality time and something I cannot imagine ever giving up. I would recommend it to anyone suffering from stress as the perfect escape from the world and the perfect opportunity for a bit of freedom and "me time".