My mother lives with us. We planned on this happening for a while and tried to do long-term budgeting for this. We also supported dh's family in Turkey for 3 years when they were having a rough time - this was about 4 years ago. You are right - it's truly all about budgeting, but it's also about cutting back, getting debt-free, and prioritizing. We used to live in Germany, and life was a little sweeter then, but we now have to face the cold harsh reality of a bozo in the White House and a sagging US economy. It's not easy to support your own family in these times, let alone another family member, but it can be done.
Assuming a house mortgage is not a debt, but an investment, we are debt-free and that has been single-handedly the most important aspect to affording to help family. No credit card debt, no car payments, no other loans. We have dd's (3 years old) college paid for with 529 investments and we invest every month in our retirement. We bought my mother a new, reliable car 5 years ago and we drive the old, less reliable cars. If we are ever in a bind, we have mom's car to fall back to. We also live in a nice house, but one that costs about 1/4 of what the bank says we can afford. I think a lot of people buy what they can "afford", but that makes them house-poor. I shop thrift when I can and we have a "food" budget that we try to stick to. We don't eat out a lot, which allows more for groceries. We try to make mom as much a part of the family as we can, but she has her debt from years ago, so we do "pay" her $500 to supplement her social security. In general, we're just careful. We also spent our early years making sure we have a savings account to fall back on and we try to keep that at around 30K or so for peace of mind. That's all I can think of now. But thinking long-term and budgeting accordingly has been a really big help.
ETA - I have two sisters who are not only furious that my mother lives with us, but has not given her a dime to help her with anything... ever. Don't count on siblings for any help. I think it's wonderful that you would want to help your sisters. I know mine would never lift a finger to help me.
