When you can save a good amount of money in your emergency fund then you can start investing your bitcoin
You don't need to set up an emergency funds first before you should start your bitcoin investment. It's better to start your bitcoin investment first with your discretionary income by sharing it into two parts and use one part for your weekly DCA and the other part to set up your emergency funds and build both of the simultaneously. Your emergency funds should reach at least three months of your income.
If you are waiting to build your emergency funds first, you are wasting a lot of time in vain piling up fiat that will depreciate in value overtime and what are you backing up when you don't have any bitcoin investment. The more you wait, the more opportunity that you will miss and bitcoin price will keep on increasing.
But I don't think you need a stable income to invest in bitcoin.
You only need your discretionary income to invest in bitcoin and not a stable income, because some people have a stable income but they don't have discretionary income.
let's say if your monthly income is $500 then you can deposit $200 in your emergency fund and with the remaining $200 you can save bitcoin which will be used to save bitcoin along with depositing money in your emergency fund. You can spend the remaining $100 for your maximum and family expenses.
You need to calculate the money for your basic needs and monthly expenses first and talke it out from the $500 and whatever is left is your discretionary income and that's what you can use to invest in bitcoin and build your emergency fund. Don't invest first without making proper calculations so that, you don't use more than your discretionary income to invest because you will end up selling the bitcoin that you bought which shouldn't be.