How much equity do you need?
Enough to keep your new loan within your loan type's cash-out LTV cap after you pay off the old balance. Because conventional and FHA cash-out generally top out around 80% LTV, you have to leave at least 20% equity in the home, which means you usually need meaningfully more than 20% equity before there is room to take any real cash out.
Those LTV caps are set by each loan program, not by this page, so I read the figure and link the relevant loan guide rather than repeat the rules. The table below shows the shape of it; your exact number depends on your home's value, your current balance, the loan type, and your file.
What is the cash-out LTV by loan type?
Each loan type sets its own maximum cash-out LTV, and that cap is what bounds how much equity you can turn into cash to pay off debt. Conventional and FHA carry a published cap; VA, jumbo, and USDA are handled by their own guides.