How can I tell my boss how stressed I am without it sounding like I'm incapable of doing my job?

Approach him/her privately and explain your situation. Let them know it wont impact your position but you felt like they should know whats going on

Let your boss know that you are feeling a little under pressure, and ask them for advice, that way you are not complaining but asking for assistance

It's important to bring this up with your boss - sometimes the stresses at your job actually come from within (aka pressure you are putting on yourself!) but other times they come from external sources, like your boss or co-workers. There are stresses that you are going to have to learn how to live with and deal with, but job related stress coming from external forces can be tamed by directly talking to the people causing it for you! Everyone deals with stress in one way or another. Don't feel ashamed to bring this up to your boss!

Part of doing a good job is self-care. Jobs that are done well are done by people that actually care about their job. Putting this much energy into a career takes energy out of you, you are not indifferent to the future of the company or the people that work in it so this takes emotional “buy in” and that can eventually take a toll. Now if your supervisors like the job you’re doing they should know not to work you into the ground, Quality work sometimes needs quality relaxation and time away to recharge. Tell them that you’re exhausted and you want to be in peak condition in order to give your Job the attention it deserves. It’s in their best interests as well.

Tell him that you need an emotional break. That it stresses you out a bit to keep up, but you can do it.

It's all about the tone in your voice, your gesture and how you bring it within communication. Just tell him you are feeling stressed lately and you wanted him to know, tell him you are still working hard and motivated, but yet you want him to know. Hopefully he will respect that fact. That's what a real boss should do.

Sometimes talking to the boss about your stress could make it where if you was less stress you could do better quality at the job. Just because you are stress doesn't always mean you are busy.

Be upfront and honest. Talk about how you are trying to go about the work and explain what exactly you are having a hard time with. Most bosses like honesty and are happy to help if asked.

Try to have a sit down with them in the manager's office and explain how you're feeling. Explain that sometimes you feel overwhelmed

I'm sorry your dealing with a lot of stress. From my experience, just being open about any issues is a pretty direct and efficient way to go about it to the boss. I've informed my manager a few times that I was dealing with a lot of personal problems, and he understood where I was coming from, even had a story to tell ontop of it. I was given what I requested, fewer hours to have more time to cope with personal issues. You never know, it can turn out better than you'd expect.