I owed a large amount to Federal taxes but got a large state refund?

It's hard to say; possibly nothing is wrong. The amount you owe on your Federal taxes depends on your total income and the amount that you had withheld towards Federal taxes or prepaid towards the Federal taxes. The state refund is based on your income (as computed by your state) and the amount of state withholding or state estimated payments. 

For example, if you had significant income, zero federal withholding and lots of state withholding, you could owe a large Federal tax balance and have a large state refund. Or perhaps if your state does not tax certain types of income, and you had large amounts of that income, you could have a state refund even though that income is Federally taxable and could create a Federal balance due. 

Carefully review each line of your Federal and state tax returns and see if any of the numbers don't make sense. That would be a good starting point.