A company or a job can't experience or "have" stress. Whatever you mean by "stress" doesn't describe a quality of a job or company or employer.
A person may experience stress working at a particular job, or for a particular company, but the stress comes from the person's reaction to the job. Every person will react differently, of course.
Some companies seem poorly-managed, creating chaos for the employees. Some managers demand grueling hours, impose poor work conditions, create or tolerate hostile environments, etc. that may cause employees to experience stress or dissatisfaction. You may hear about those companies anecdotally, or read about the chaos or the layoffs and lawsuits here on HN. Sites such as Glassdoor may have useful information about the conditions and management style at a prospective job.
I suggest not making the work you do or the company that pays you part of your identity. One way employers cause stress starts with getting employees to identify with the company, calling it a family, imposing rituals and in-group language. That can cause employees to feel shame and guilt when they want to go home on time, not work on weekends, not go on call, take a vacation, even getting sick. If you want to separate personal life from work and stay sane, as you put it, set your boundaries and don't let the job violate them.
