In English, clausal arguments of a verb pattern with indirect questions and contrast with relative clauses:

Inanimate referent

What annoys you? (Direct question)

I wonder what annoys you (Indirect question)

The thing that/which annoys you (Relative clause bound to a noun)

I brought what annoys you ("Free" -that is to say unbound to a noun- relative clause that serves as the object of brought)

Animate referent

Who annoys you? (Direct question)

I wonder who annoys you (Indirect question)

The person that/who annoys you (Relative clause bound to a noun)

I brought the person who annoys you (a free relative clause is impossible here)

In French, things are slightly more complicated, but follow a similar logic:

Inanimate referent

Qu’est-ce qui t’ennuies ? (Direct question)

Je me demande ce qui t’ennuies (Indirect question)

La chose qui t’ennuies (Relative clause bound to a noun)

J’ai amené ce qui t’ennuies (Free relative clause)

Animate referent

Qui est-ce qui t’ennuies ? (Direct question)

Je me demande qui t’ennuies (Indirect question)

La personne qui t’ennuies (Relative clause bound to a noun)

J’ai amené la personne/celui qui t’ennuies (a free relative clause is impossible here)

Determining the exact function of "ce" in such clauses is more complicated and has made a lot of ink spill. Giving an exact answer would require a specific theory of syntax, which would be unhelpful here and lead the answer into theoretical linguistics.

Here’s what we can tell:

• "ce+relative pronoun" appeared in Old French (with older forms of ce, like cil, cist, etc. that were equivalent to modern celui-ci/-là, celle-ci/-là), but was almost completely limited to relative clauses. In indirect questions, only que and qui were used, without a pronoun. It’s only in the 17th century that the forms with ce take over.

• Older stages French could sometimes directly use its relative pronouns in an argument position of a verb or a preposition (qui casse paie ((the one) who breaks (it) pays (for it)), a saying using an older syntax, or administrative Belgian French "ce poste est ouvert à tout qui veut" (this position is open to all who wants it)), but started needing to bring in a dummy noun or pronoun to act like the object and have the relative clause act like its complement. The comparison to "that which" in the OP is apt (or "here where" or "him who", that also occurs in English but have fallen out of colloquial usage).

• It’s probably significant that other démonstrative pronoun + relative pronouns combinations occur in free relative contexts: "Je viens de là où le soleil se couche" (I come from (there) where the sun sets), "J’ai réparé celui qui est tombé" (I fixed the one that fell), but that they can’t occur in indirect questions: "Je me demande de (*là) où tu viens", "Je me demande *celui qui a cassé en premier".

• It’s clear that question words and relative pronouns share some similarities and that either one can grammaticalise into the other through the same path "ce+relative pronoun" took: in Portuguese, an equivalent construction (o que = le que) has become the normal word for what, even in direct questions (O que quer o senhor? = what does the sir want?) and "qu’est ce que" (and variants) is taking the opposite route in colloquial French, although this usage is still stigmatised (Qu’est ce qu’il veux ? -> Je me demande qu’est-ce qu’il veut -> #J’ai appris qu’est-ce qu’il veut). English takes it one step further with what, that can serve as a nominal relative in stigmatised use: "#The man what we saw"

• So "ce+relative pronoun" saw a change in status that allowed it to spread where other demonstrative+relative combos couldn’t go and eventually to become obligatory.

• Characterising this change in a theory neutral way is beyond my abilities. At the very least, it’s clear that ce not a full question word (since it can’t be used in direct questions) nor a simple demonstrative since it does what other demonstrative pronouns can’t do.

• For now, just accept that the morpheme ce (that has a very different syntactic behaviour from subject pronoun ce or determiner ce and can be considered fully distinct from them) carries some grammatical information that allows its host to serve as the complement of a verb and that confers some question-word-ness to that same host (but not fully, since "ce que" and the like can’t serve as a question word in direct questions). In this it is similar to "what" in English. Meanwhile, the relative pronoun part of the construction marks case, either intrinsically or through a preposition+quoi:

Je veux ce que tu as acheté – I want what you bought

Je veux ce qui t’intéresse – I want what interests you

Je veux ce dont tu parles – I want what you’re talking about

Je veux ce sur quoi tu es assis – I want what you’re sitting on

Je veux ce à quoi tu tiens – I want what you hold dear (=what you’re holding on to)

In French, if you want to translate "I don’t know what to do." you will not say preferentially "Je ne sais pas ce que je fais." (que" is a relative pronoun) but you say as well "Je ne sais pas quoi faire." and that means as in English "I don’t know what I will do.". You’ll notice that in this form involving "quoi", the word "ce" is not used; "quoi" is not a relative pronoun but an interrogative pronoun used in an indirect question just as in "I don’t know what to do." ("what" is an interrogative pronoun in an indirect question). When the form "Je ne sais pas ce que fais." is used that way, the tense is not "présent" but "présent à valeur de futur proche".

Neverheless, in using this form involving "ce" (added "ce", a demonstrative) you imply a context your locutor is aware of, a context in which what you might do has been discussed, envisaged. Thus, "ce" has the fonction of referring to this thing; in fact you could say "Je ne sais pas la chose que je fais." and the neuter pronoun "ce" is nothing but a systematised way (linguistic systematisation) of referring to something that could be called a thing, a matter, a business and what have you. In English, this idea has been gathered under the word "what", which, in this context, is not an interrogative pronoun in traditional English grammar but a relative pronoun, as in French. By means of such a construction ("ce que") you satisfy to the necessity, in connecting "savoir" to "faire", of providing an object to "savoir" and an antecedent to "quoi".

A second possibility for this form ("Je ne sais pas ce que je fais.") is the case where the tense is decoded as a "présent" and then your English translation is surely "I don’t know what I’m doing.". In French "ce" is added because you need an object for the verb "savoir" (as in the first possibility) and as well an antecedent for the relative pronoun (qui, que, quoi, dont). The outcome of this is that there is no stated antecedent for "what" in English; as well, there is no stated referent for the pronoun "ce" in French. In this second case, the antecedent is again to be found in the background that unite the locutor to the person spoken to (for instance, a task the locutor is performing usually but that he can’t perform well any more, a recent discussion in which the locutor had mentioned that he/she was slowly becoming mentally ill, …)

In English you say "What I do is my business.", for instance. In French the translation is "Ce que je fais me regarde.". You will perhaps understand better when you find out in the explanations below that in English too you sometimes have to rely on a demonstrative (that); it just happens that in English usually a simplified form is used and that this construction is not often needed.

The forms "CE QUI", "CE QUE", "CE QU’" (mere elision of e), "CE DONT", "CE À QUOI", are sometimes treated as pronouns (ref.) and you might consider this option for the sake of simplification. You can use the following correspondence. However, there might be other forms (supplement or alternatives) as I made up the list given from memory. Nevertheless, in traditional grammar "ce" is a pronoun which has for function "antécédent du pronom relatif « qui », « que », « dont », ou « quoi »"

• ce qui        => what
• ce que       => what
• ce à quoi   => what, that which
• ce dont      => what

1.sujet (subject case) : ce qui

• Ce qui me fatigue n’est pas le travail. → What is tiring me is not work.

2.objet (object case) : ce que

• Ce que vous dites est vrai. → What you are saying is true

3."objet indirect « à »" : ce à quoi (penser à, arriver à, contibuer à,… depends on the prep. used with the verb)

• Ce à quoi vous pensez est autre chose.
  → (usual) What you are thinking about is something else.
  → That which you are thinking about is something else. (This construction, although it is still found, is not often used.)

• Ce à quoi vous êtes arrivé est un succès. → What you’ve managed to accomplish is a success. (In English, the French "indirect object" becomes an object, therefore you use "what".)

• Ce à quoi vous avez contribué est le progrès dans l’enseignement.
  → (usual) What you have contributed to is progress in teaching.
  → That which you have contibuted to is progress in teaching.
  (alternative) That to which you have contributed is progress in teaching.

4."objet indirect « de »" (parler de, se soucier de, traiter de,…depends on the prep. used with the verb)

• Ce dont vous parlez est la foire internationale. → What you are talking about is the international fair.
• Ce dont vous vous souciez c’est (de) vous. → What you care about is yourself.
• Ce dont vous aurez à traiter est la première proposition. → What you’ll have to explain is the first proposition.

The Guide of the Perplexed – Volume 1 – Page 67
https://books.google.fr › books 2010

• For man has in his nature a love of, and an inclination for, that to which he is habituated.

• (A more common alternative) For man has in his nature a love of, and an inclination for what he is habituated to.

• C’est ainsi car l’homme, par nature, a un amour et une inclination pour ce à quoi il est habitué.

The Encyclopedia of Philosophy – Page 243
https://books.google.fr › books 2012

• One meaning of "universe" is "that to which the observed universe belongs"; another is "that which is characterized by a cosmological model."

• Une signification du mot « univers » est « ce à quoi l’univers observable appartient » ; une autre est « ce qui est caractérisé par un modèle cosmologique. ».

Philosophy Classics: Greek and Roman Philosophers (Includes …
https://books.google.fr › books 1967

• Consequently if again he thinks that A belongs to nothing to which C belongs, he thinks that A does not belong to some of that to which B belongs;

• En conséquence, si alternativement il pense que A n’appartient à rien de ce qui appartient à C, il pense que A n’appartient pas à quoi que ce soit de ce à quoi appartient B ;

In addition to the excellent Eau qui dort’s answer, note that "Je ne sais pas que je fais" is not technically incorrect. It just means something else: "I don’t known that I do"

On the other hand, there is a trend in very relaxed colloquial French to say:

Je ne sais pas quoi je fais.

It matches your expectations but be aware that it is still considered very incorrect despite being understood by everyone.