Mdthbs

I would use the term independently wealthy, though I don’t think it’s considered an idiom:

(of a person) Possessing enough wealth that one does not need financial support from another person and does not require income from employment.

-YourDictionary

The phrase set for life was the first that came to my mind. It doesn't necessarily imply making a lot of money, but with some additional words you can make some idiomatic phrases.

If you make it big on Broadway, you'll be set for life.

You'd be set for life if you'd invested in the month after their IPO.

Another possibility is to strike it rich. This usually but not always means a person is set for life, but rather that they made a lot of money doing one thing or another.

By the Great Horn Spoon! uses this phrase a lot in the context of people migrating to mine gold. Example from the summary (likely many more in the book text):

They pick a beautiful spot, but then, as they hit bedrock, Jack and Praiseworthy find gold and strike it rich.

How these two relate to other answers suggestions:

• Anecdotally, make a pile is rarer than other suggestions here, even when spoken. The Ngram supports this notion, but make a pile's search results (18.5 million at time of writing) are greater than strike it rich's at 1.2 million. Then again, "make a pile" can refer to making a physical pile of objects, so it's hard to tell how many of these results are being used in the context this question is asking about.




• make it big does usually refer to money, but "making it" can refer to a more general notion of success, with "big" simply amplifying it




• Here's sources for make a fortune versus make one's fortune. Making a fortune simply refer to making a lot of money. In context, this could be a smaller amount than needed to be set for life. On the other hand,
  "one's fortune" is all the money someone needs to be set for life.

Ngram viewer with a few of these phrases

It's possible none of these phrases have the exact connotations and usage you're looking for. But if you combine them together and give enough context, you can build your own idiomatic phrases.

I can't address the idiom "make a pile" as I am not familiar with it, but maybe some else can. Here's what came to mind for me:

• make it (big)
  
  infml to become famous or successful:
  By the time he was nineteen, he had made it big in the music business.
  
  (Cambridge Dictionary)



• 
  

  make it big

  
  
  
  
  
  • To achieve great success and/or fame.
    My dream is to make it big on Broadway.
    You're never going to make it big if you don't put in the hard work.
    
  
  
  • to become successful, especially financially. I always knew that someday I would make it big. My brother made it big, but it has just led to tax problems.
    
    (TFD)
  
  
  
  

However, this also doesn't necessarily mean they're no longer in need of money and retire. In any case, "... and after a while he made it big selling silk carpets and retired by the time he was 40" sounds like a perfectly idiomatic sequence to me.

Anyway, there might be an idiom out there that captures exactly what you want. Let's see!

As you say, make a pile simply means that you have made, or are making, a lot of money: it doesn't carry any implications that you don't need any more money.

Once you have reached the stage where earning any more wouldn't make any difference to you, you could use the informal expression filthy rich.

I'd use the phrase to make one's fortune, which means to become wealthy.

(None of the references spell it out, but I've always heard it in reference to making enough that money is no longer a concern.  Of course, people who make a fortune through their own efforts are rarely the sort to stop working at that point, but I'd certainly assume that to be an option.)

"Financially independent" is a common term for this. Specifically, it means that you don't depend on anyone else for your money, because you have enough money yourself.

The term is sometimes encountered in the idiom "Financial Independence Retirement Early" (FIRE). That seems to match the exact example you gave, so: ""He achieved financial independence selling silk carpets and retired by the time he was 40.

F*** You Rich is one used in internet circles.
The definition can be described as "So wealthy that you can say F*** You with no cause for concern or fear of financial reprisal".

The idiom that works here is:

make a fortune. In this case, "made a fortune".

It's that simple.

I would suggest 'Rolling in it' as an alternative.

Used in the example context:

As a sociable person he managed to connect to the market's most influential people and the big shots in his own field. He was rolling in it by the time he was 40, after selling carpets, and decided to retire.

When used in the context of money it means to have so much money you can afford to lie around all day surrounded by your money.

It also has the connotation of having a lot more money than you need - this can apply to the very wealthy or can also be used in other contexts if someone has more money that expected.

The phrase "Independently Wealthy" comes to mind.

after awhile he ... Became Independently Wealthy... selling silk carpets

Wictionary says:
https://en.m.wiktionary.org/wiki/independently_wealthy

Adjective

independently wealthy

Possessing enough wealth that one does not need financial support from others or income from employment.