Tout simplement :

Je suis malade et je vais me reposer à la maison.

The second guess does it: “je vais me reposer à la maison.”. However, instead of “à la maison” you can say “chez moi” if you speak to people you don’t know well, the more formal register of “chez moi” is preferable.

There are two reposer verbs in French. The first one means re-poser, i.e. poser ne nouveau, and the second one relates to repos and pause (break) and used to be written repauser in early French.

The first one is always transitive so is excluded in the first and third attempts.

Examples of its usage can be:

Je vais reposer du carrelage à la maison.

I’m going to lay tiles again in my home.

Je vais reposer la question à mon voisin.

I’m going to ask the question again to my neighbor.

The second reposer verb matches rest.

Je repose à la maison

This used to be possible in the past but is no more used in modern French, outside archaic regional usage. It might still be heard in Provence for example. Otherwise, it would have the unlikely meaning:

I rest (in peace) at home (i.e. I’m dead.)

Here is a non reflexive way to use reposer that has a meaning closer to your English sentence:

Je vais reposer mes yeux à la maison:

I’m going to rest my eyes at home.

Je vais me reposer à la maison:

I’m going to have a rest at home.