You do not have to be flexible to do yoga - flexibility will come from the practice of yoga and time. Anyone can practice yoga, if they have the right attitude, and/or the right teacher. Ideally, you can find the perfect teacher and class for you that meets your needs. Some classes have students sit in chairs who might otherwise not be able to rise from the floor.
When you are in a pose, it should feel comfortable and pleasant. Boring is fine. Pleasure is even better. Discomfort is not an option. If a pose is unpleasant, and you persist, you are no longer practicing yoga. You have crossed over into the world of self abuse. Ease up and relax and do not worry about the outward form of the pose. Modify your position in such a way that it is easy and comfortable, and then go just a little deeper. Be patient, and allow yoga to do its magic over a period of months and years.
Try breathing into the area of your body being hardest worked in any pose. As you breath out, turn your awareness inward. Yoga is a feeling, a state of mind, Not simply the physical pose.
Personal progress should never be considered. Do not compare yourself to others. Do not even compare yourself to your own past performances, or expectations. Any comparison is an unnecessary and dangerous distraction. It is best not to think in terms of how far you are stretching, or twisting, or how long you are holding; simply be aware of your present capacity, and explore the boundaries of your present limitations.
Focus all of your energy into learning to enjoy the Asana itself so that you practice Yoga for the joy of practicing Yoga, and not for some imagined goal, whether it be health, relaxation, or whatever, they are all distractions that inhibit the practice of Yoga and thus interfere with their own fulfillment. Let them go! Surrender to each pose in turn. Concentrate on relaxing the muscles holding you back, not tightening those pulling you further into the pose. Move slowly, allowing the body time to adjust and stretch. If it feels good hold it, explore it, get as much out of it as possible, and then move on.