I don't think it's a bad reflection on you as an adult, it's just that your family is demanding something from you that's unreasonable.
There's more than one way to respond. You could refuse outright to cosign for the loan. Another approach might be to cosign for an amount that is less than they want, but within what you could afford to pay if this all ended up getting dumped on you.
Your family might be unhappy if you went with one of the options above, but it's also ok if they're unhappy, as long as you could deal with an repercussions from that.
If you decide to cosign for the loan that they want, there could be cost overruns, especially if the contractor is unscrupulous. Also, even if everything goes as planned, the fact that the bank doesn't want to loan them the money without a cosigner indicates that they're not creditworthy, so they think there's too high a risk they won't pay the money back.
Have you ever cosigned on a loan for them before?