Can languages with infinite strings be recursively enumerable? I am not 100% sure about the definition of recursively enumarable languages. Yes I know how are they defined: There has to exist a Turing machine that accepts all wrods of the language and halts but it does not have to halt on different inputs.

But suppose I have a language consisting of uncountable size with words of infinite length. Will Turing machine halt with a word of infinite length?

Consider a language constisting of all binary strings (I can produce such a set by diagonalization - Cantors rule, for example). Is this language recursively enumarable? If so, can you tell me why? I have not found any definition of this that would explain the behavior of TM with such inputs...
As you state, the standard definition of TM acceptance requires that the TM halts after finite time.

It is still possible for a TM to accept languages that contain infinite words, if they can accept without reading the entire word. For example the language of all (finite and infinite) words that contain at least one $0$ is semi-decidable, since if there is a $0$, the TM will encounter it after a finite amount of time and can then accept the word.
The complement of this language however is not semi-decidable, since it is not possible to read an infinite input in finite time (and it is not possible, in the general case, to know that there is no $0$, without reading the whole input).