I will answer your question simply. Is it possible to work for a boss who is critical of everything you do? Yes, it is possible. What you need to determine is, is it worth it? A great variable here is the method in which criticism is delivered. If it is delivered in a manner that allows you to learn and grow professionally, then the criticism would have more a more positive effect. If a boss likes to berate employees and belittle them I would not think that would be worth tolerating.
If you decide that you can deal with a boss criticizing everything you do, you will need to be able to manage your own emotions and learn not to take criticism or feedback on performance negatively. After all, it is a job. An employer has a right to ask that a job be done in a certain way. So, you need to be able to separate criticism on performance from your value as a person. Learn to understand your employer's expectations.
That being said, if you are unfortunate and have a boss that likes to criticize everything you do just to hear his/her own voice, you need to determine what's in it for you? Is the job you have providing income and benefits to offset the negative criticism? Are you able to listen the criticism and maintain your value as a person? There are some who are very successful with this. In this time of high unemployment, there are probably more people in this situation than any of us would like to think about.
One final thought is we all learn by experience. So, the boss who provides feedback in a positive way that allows you to grow was most likely mentored by another leader who trained him/her in the same way. By the same token, the boss who berates and belittles may also be following in the footsteps of the one who managed him/her. Only leaders who carefully consider who they want to be and how they want to lead can get themselves out of this trap.