It sounds like your current loan is in your name. As such, you are responsible for paying it. Not your family, you. It also sounds like the loan payments are regularly late. That'll likely drastically affect your credit rating.
Given what you've said, it doesn't surprise me that you were declined for a credit card. With the information on your credit report, you are a poor risk.
Assuming your family is unable to pay loan on time (and assuming you aren't willing to do so), you desperately need to get your name off the loan. This may mean selling the property and closing out the loan. This won't be enough to fix your credit, though. All that will do is stop making your credit worse. It'll take a few years (five years in Canada, not sure how many years in India) until this loan stops showing up on your credit report. That's why it is important to do this immediately.
Now, can a bank give you a loan or a credit card despite bad credit? Yes, absolutely. It all depends on how bad your credit is. If the bank is willing to do so, they'll most likely charge a higher interest rate. But the bank may well decide not to give you a loan. After all, your credit report shows you don't make your loan payments on time.
You may also want to request your own copy of your credit report. You may have to pay for this, especially if you want to see your score. This could be valuable information if you are looking to fix your finances, and may be worth the cost. If you are sure it's just this one loan, it may not be necessary.
In India CIBIL is the authority that maintains records. Getting to know you exact score will help. CIBIL offers it via TransUnion.
The non-payment will keep appearing on your record for 3 years.
As you don't have any loans, get a credit card from a Bank where you have Fixed Deposits / PPF Account as it would be easier to get one. It can then help you build the credit.