First, let me nitpick about another topic:
this "no better man", that looks strange. I've seen it in phrases like i.e. "there's no better man for the job", but what's the meaning here exactly?
This is an older phrase, when the word "him" could often be seen when referring to people of both genders, or an unidentified person who might be a female. This phrase may also have come from the time before women were commonly a part of America's workforce. So, don't get worried about the term "man". This is equivalent to:
there's no better person to do this task
If a person is the best person for the task, then nobody is better than this person. If the person does a mediocre job, then there might be many other people who would be able to do the task better.
Now, let's move onto the main topic:
can those phrases be considered correct [E]nglish...?
I'm answering from my perspective of American English. There is a famous phrase:
“Ain’t” ain’t a word.
For a long time, the word “ain’t” has been relegated as a slang word. The word was considered absolutely inappropriate to use in certain circumstances, such as any formal usage including any school assignment for a child. Most dictionaries did not bother including the word. (Newer dictionaries, particularly online dictionaries which can be quite large without the expense of printed paper, may be more prone to show the word, so that a spelling can be identified and a definition can be provided.)
Wikipedia's article for “ain’t” says, “The usage of ain't is a perennial subject of controversy in English. Ain't is commonly used by many speakers in oral or informal settings, especially in certain regions and dialects. Its usage is often highly stigmatized, and it may be used” to identify people are unfamiliar with proper speech, which may indicate they are less educated.
In the spirit of the idea that “ain’t ain’t a word”, the song is intentionally being a bit unclear by choosing the include the word. Generally speech is intended to communicate, but in this case, the song is sacrificing some clarity, and is doing that for the sake of having a bit of fun. Many times, the primary goal of a song is not to communicate, but to have people experience enjoyment. So, having some fun may be appropriate.
“This ain’t no place for no hero”
The phrase “no hero” can rightfully be replaced with the phrase “coward”. I am going to so some variations of the sentence, to make an analysis be more clear.
“This ain’t no place for a coward”
Maybe a hero would stand in a spotlight, honored for the heroic deeds. Another place of honor might be on top of a podium, where winners are usually honored. Lots of people can see whoever stands there. Maybe there is a “heroes hall”. Such places are appropriate for heroes to walk down. Such a place is not appropriate for unworthy people like cowards. Such a place might not be a place intended for cowards. Such a place might be “no place for a coward”. Instead, the place is a good spot for a hero.
“This ain’t a good place for a hero”
However, “this place” might be a shelter, such as a small cave accessed by a hidden hole in the ground. Most people would not even know about it. Such a place would be a perfect place for a coward (not a hero).
So, there you have an analysis of breaking down the phrase, piece by piece. It looks like the simpler version of the phrase is saying:
This is no place for heroes.
That makes sense: two of the negatives essentially end up canceling each other out, and so you end up with a remaining negative.
The phrase seems to be saying that the place is perfect for cowards.
We're using a triple negative, which is exceedingly rare in formal English. So we are probably looking at an extremely challenging concept, or we are looking at someone who is violating standard rules. Since the word “ain’t” is being used, we are probably looking at someone breaking the typical rules.
Sometimes, people use multiple negatives to really reinforce the point that what they have in mind is negative. I personally disagree with the statement from Hellion's answer which says:
Most of the time, especially in "vernacular" speech, multiple negations are not intended to be interpreted sequentially, but rather as an intensified single negative.
I disagree with the assessment that this is what happens “most of the time”. However, I am basing that on opinion. I can definitely agree that I have seen people speak that way, particularly when the person is so wrapped up in emotion that they don't choose their words carefully.
Actual Idea #1
So, my initial opinion (as an experienced native speaker), what is really trying to be communicated is something more like:
“This is NO place for someone who is not a hero.”
In other words: You can be in this place. However, you're broken. You're acting like a coward. You need to fix this problem. Act heroic. Then you will deserve to be in this place.
Stop acting so embarrassing. Stop being scared. Have confidence. Be brave. Be impressive. Be worthy of the title “hero”. Be heroic.
Edit History: I added to my answer, by adding this other idea (and related analysis near the end).
That idea, which was just mentioned, was just my initial thought. However, as I thought about this further, I thought that the person might have been trying to communicate a different idea.
Actual Idea #2
The idea, which was actually trying to be communicated, might have been this: be normal. People who are recognized as heroes are often noticed for doing something unusually heroic. Stop trying to be a hero. Stop trying to be extra special. Just do things the normal way, which is the safe way, and the right away. Your attempts to be noticed, by being unusual, are not appreciated here.
w[h]ich doesn't really seem to make a lot of sense.
I personally find the communication attempt to be a bit unclear. It makes me need to think for a moment about what is really being said. That is really unfair, because the reason I need to think so hard is because the speaker did not think enough to choose words more carefully... unless the speaker wanted to make me to think about what is being said. Forcing me to slow down and think might have been the act of a genius who understands what effect the unclarity would have on me.
When I think about the situation, I can probably think about whether he is encouraging me to think of the place as a place for cowards or a place for heroes. The person is probably really trying to say whatever actually makes sense in the situation.
Note that I mentioned two possible actual ideas, and they are really saying opposite things: “be a hero”, or “stop trying to be a hero”. How can we know which idea is intended? The answer would be to pay attention to some other available clues. I haven't listened to the song, but other words might let us know what idea the speaker is actually trying to say.
Since this is actually being said as part of a song, informal speech (which violates formal rules of speech) may be very appropriate for this type of informal, fun communication.