Using email bouncebacks is the most socially acceptable “no” there is. It doesn't mean you're telling them that you don't want to respond to them. It only states that you can't reply for a certain period of time.

Using E-mail Bouncebacks

It gives you the time to pause and reflect and ultimately reply that if you are or are not available. It enables you to take back control of your own decisions rather than be rushed into a “yes” when you are asked.

“Let me check my calendar and get back to you” 

Saying "no" with humor will lessen the awkwardness and will lighten up the atmosphere.

Saying "No" With Humor

"The difference between successful people and really successful people is that really successful people say no to almost everything."

Saying no to a senior leader at work is almost unthinkable, even laughable, for many people. However, when saying yes is going to compromise your ability to make the highest level of contribution to your work, it is also your obligation. In this case it is not only reasonable to say no, it is essential. One effective way to do that is to remind your superiors what you would be neglecting if you said yes and force them to grapple with the trade-off.

“Yes. What should I deprioritize?”

Use awkward pauses as a tool to say no. When a request comes to you (this works only in person), just pause for a moment. Count to three before delivering your verdict.

The Awkward Pause

This is a particularly good way to navigate a request you would like to support somewhat but cannot throw your full weight behind. I particularly like this construct because it also expresses a respect for the other person's ability to choose, as well as your own. It reminds both parties of the choices they have.

“You are welcome to X. I am willing to Y”

“Half of the troubles of this life can be traced to saying yes too quickly and not saying no soon enough.”

It is tempting to think that our help is uniquely invaluable, but often people requesting something don't really care if we're the ones who help them- as long as they get the help.

 “I can’t do it, but X might be interested”

"Saying no is its own leadership capability. It is not just a peripheral skill. As with any ability, we start with limited experience."

E-mail is also a good way to start practicing saying "no but" because it gives you the chance to draft and redraft your "no" to make it as graceful as possible. Plus, many people find that the distance of e-mail reduces the fear of awkwardness.

The Soft "No" ("No, But")

In a world of more requests than we can possibly fulfill, learning how to say no with grace and style is a skill we all need. We should be saying no more than we say yes, although the opposite is usually true. We say yes too quickly and no too slowly.

JOSH BILLINGS

Similar ideas

You need to learn when to say both “yes” and “no.”

Feelings of Resentment

Learn to say “NO”