Learn Hardcore Korean: dongari hwaldongeul hageona gongbureul haedo dwaeyo. - You can either participate in club activities or study.

-거나 is a verb connective meaning “or.” You attach it to the present-tense stem of a verb (here 하 from 하다) to list alternative actions. So 하 + 거나 = 하거나, giving “do (club activities) or ….”

Why do we need 하다 after 동아리 활동? Can’t we just say 동아리 활동거나?

동아리 활동 is a noun (“club activities”). To turn it into an action, Korean uses the verb 하다 (“to do”). Since -거나 attaches to verb stems, you must say 동아리 활동 + 을 + 하 + 거나 = 동아리 활동을 하거나.

Why do we use the object particle 을/를 with 동아리 활동 and 공부?

In Korean, when a noun is the direct object of a verb, you mark it with 을 (after a consonant) or 를 (after a vowel). Both 동아리 활동 and 공부 are objects of 하다 here, so they become 동아리 활동을 and 공부를.

How do we form -아도 돼요, and what does it mean?

-아도/어도 돼요 attaches to a verb stem to express permission: “may do X” or “it’s OK to do X.” Drop -다 from the dictionary form, then add -아도 되다 (after bright vowels) or -어도 되다 (after dark vowels).
Example: 공부하다 → 공부하 + 어도 되다 = 공부해도 돼요 (“You may study.”)

Why isn’t 하다 (the first verb) also followed by -아도 돼요?

The structure A거나 B해도 돼요 means “You may do A or B.” Only the second verb takes -아도 돼요; the first is simply linked with -거나. You don’t repeat the permission ending after the first action.

Can I use 나 instead of 거나 for “or”?

No. To connect verbs as options, use -거나.

-나 after nouns gives alternatives (e.g., 사과나 배 = “an apple or a pear”).
-나 after verb stems can form rhetorical questions (e.g., 먹나 마시나? = “wonder if [they] eat or drink?”), but it doesn’t list simple choices.

Are there other ways to grant permission for two actions?

Yes. You can make each verb carry the permission ending:

동아리 활동을 해도 되고, 공부를 해도 돼요.
Or use a more formal ending:
동아리 활동을 해도 되고, 공부를 해도 됩니다.
You can also use -든지 for “either… or…”:
동아리 활동을 하든지, 공부를 하든지 괜찮아요.