So, I'm thinking about that, is the JTextArea length infinite? Or how many chars can be used max?
No, JTextArea is not infinite.
We can imply the maximum length based on the fact that JTextArea only returns a String, which has a length which returns a int. This implies that the maximum length of a JTextArea is bound to Integer.MAX_VALUE, but, because of array overheads, is slightly smaller. But in practice, you'll probably find that it's much smaller, due to the need for arrays to be laid in memory in a continuous manner, so it will depend on how much memory the JVM has available and how fragmented it is.
We can further investigate this and have a look at PlainDocument, which is the default Document used by JTextArea, which uses a char[] as it's internal data structure, just like String.
This further concretes the reasoning that the limit of a JTextArea is limited to less then Integer.MAX_VALUE
You can have a look at Do Java arrays have a maximum size?, Why I can't create an array with large size? and Why the maximum array size of ArrayList is Integer.MAX_VALUE - 8? for discussions on why an array can't be declared as Integer.MAX_VALUE
Now, before someone suggests that you could write a linked list implementation of a Document, don't forget that both Document and JTextArea rely on String, which is a key limiting factor