Too many commas! Omit commas after "house", "problem". "Saturday" (capitalized) "been unemployed" "under wrapS with her"

Sounds fine. You just need to add the verb ‘to be’ when you say ‘unemployed’. I’d lost my job and have been unemployed for five months

Natural is: On Saturday, Luna visited my house and asked me how my work was. I told her that it was fine, but, truth be told, I’d lost my job and been unemployed for 5 months. I didn’t want to upset her. In the US, the expression is ‘keep something under wraps’ (not ‘wrap’). It doesn’t really apply here because it suggests temporarily withholding information. It’s unnecessary and therefore unnatural.