Is there such a thing as a hardship enlistment, specifically for the US Navy. If a person who is drug and alcohol free, no criminal record?

a what?  no.  there is not.  you either meet the standards for enlistment, or you do not.  there are some other things in the mix too...criminal record, credit check, but basically that all falls under the 'meet the standards' objective.  there is no 'right' to serve in the military.  they do not have to take all applicants.

you really were not very descriptive as to why you are asking.  so what is going on?

Even though, the military seems like a social welfare program, and it is also used by the politicians for their social experiment sometimes, it still does not have a Hardship Enlistment only because you needed a job. 

Not really. You must already be in DEP and have your circumstances drastically change

NO. A hardship DISCHARGE exists. If you don't meet the requirements to enlists, a hardship would make you even LESS qualified. There are cases where it is possible to get a waiver for some things that would normally disqualify you if there is need for more people in the job you go into, but that isn't a hardship.

Not that I ever heard of. Why would a person need a hardship enlistment if they're drug and alcohol free with no criminal record ?

There is no such thing as a "hardship enlistment".  You either meet the standards or you don't.

There is no "right to serve" or requirement that the military must take you.

Drug free, high school diploma, able to pass the ASVAB test with a qualifying score, and meeting the height/weight requirements are VERY easy, but many young people can't even meet those minimum screening criteria.

there might be, i would check that out and see

If you're asking if they would lower the standards for someone who is poor, then no.

Why not just visit a recruiter and see if you qualify on your merits?