We all care about others’ feelings a lot, especially when we say NO. Typically, there are three ways to say NO indirectly/politely.
(1) 생각해볼게요 (I’ll think about it)
This phrase gives an impression that you’re taking the other person’s offer seriously instead of rejecting it right away. If you want to be even more polite, you could specify how much more time you need.
(2) 미안해. 사정이 생겨서 못할 것 같아. (I’m sorry. Something came up)
This is another common phrase to say NO. ‘사정’ sounds a little private and serious, so the other person might not dare to ask exactly what happened to you unless he/she is very close to you.
(3) 미안해. 급한 일이 생겨서 못할 것 같아. (I’m sorry. Something urgent came up)
This means basicaly the same with “미안해. 사정이 생겨서 못할 것 같아.” But since you used “일” in this phrase, the other might ask exactly what happened.
Please keep in mind that you cannot overuse those phrases, because it will eventually give others have a bad impression on you.