Make sure he is not ill or in pain. Most babies start teething around 6 months old, however, some babies can begin teething sooner. One of my children developed a tooth earlier than average. Examine his gums and see if they are puffy or swollen. If it looks like he could possibly be getting a tooth, you can treat him with over the counter infant pain reliever as well as a topical oral numbing gel for infants.
Could he have colic? Sometimes the pain and pressure from colic is relieved when the baby is held upright, so this may be one of the reasons he wants to be held. Symptoms of colic can include tightness in the tummy, clenched fists, pulling the legs toward the tummy, lots of facial redness during crying, and passing gas. If he could be suffering from colic, warm compress on the tummy can help, as well as gentle massages. If you use a warm compress, never use a heating pad, as they can get too hot for an infant. Use a warm water bottle. Unfortunately, there aren't a vast variety of options when it comes to colic other than keeping him comfortable. You could try gas relief drops that are made for babies. Also make sure he is burping good after feedings.
Make sure he is not sick. Check his temperature. If he has a fever, vomiting, not passing normal stools or passing no stools at all, or not passing urine, take him to his pediatrician immediately. If the pediatrician is out of the office, take him to the emergency room. Always better safe than sorry.
If it does not appear that he is in pain, sick, in need of a diaper change, or hungry, it can be very frustrating. When my babies cried, I held them as much as I possibly could, but of course, it is impossible to hold them every waking minute, even when they are crying. There were also times when I needed a break. I never let my babies cry for prolonged periods of time, but there were certainly times when I need a little break, and had to walk away for a bit. If you are not able to hold him or if you simply need a break, put him in his crib face up, and make sure there is nothing else in his crib such as blankets, pillows, toys, etc. While it is very difficult to leave him when he cries, remember, babies cry. It is completely normal. Crying will not hurt him at all. Check on him periodically to make sure he is okay. There is nothing wrong with having to take a break and walk away for a bit. You cannot be the best mom you can be if you are in a constant state of anxiety, frustration, exhaustion, and on the verge of a breakdown. Don't feel guilty about taking a break or even calling a close friend and/or family member to come and take over for a bit. If no one is available to help you, take a break and don't feel guilty about doing it. As long as he is healthy and safe, he will be fine.