How can you reenlist with a RE code 4?

You have to talk to a recruiter, and try to get a waiver. These days, you won't get a waiver, the military is downsizing big time.

You can apply to have your RE code changed at the Board for Correction of Naval Records or the Board for Correction of Military Records depeding on which service you were in. The services' discharge review boards do not have the authority to change RE codes. You will have to show an administrative error, or some injustice in order to make a change. RE codes are very rarely changed. But, if you are successful, you may then be able to re-enlist.