I got an offer from capital one for a credit card. It said I was preapproved...blah blah blah. I wanted to continue to build my credit but I also needed it as an emergency back up in case I ever have to leave the state for a family emergency. It did not mention the credit limit but I applied for it anyway and was approved. Today, I got it in the mail and it only has a 300 dollar credit limit which won't do me any good whatsoever. I have to activate the card before I can use it. I don't want it now so can I just not activate it? Can I tell them I don't want it? And most important....Will turning it down, hurt my credit now?
It carrys a 45 dollar annual fee.

Depends. If you have to pay annual fees, then tell them that you don't want it. If no annual fees, then keep it. My economics teacher used to tell me that it is good to apply as many credit card as possible, since you will never know when you need extra money for emergency, etc. Of course, as I said, keep only those that don't have annual fees.

If you want to build up your credit, then taking the card is probably a good idea, if you are responsible.
I have a bunch of credit cards and I use them all the time, only thing is I try always not to have an outstanding balance that will earn interest and/or late fees for the credit card company.
By using the card with the $300 limit and not paying late and paying off the balance each billing cycle you will eventually have your limit increased.
Get a gas credit card too, just use it for gas and manage your money to pay it off every month.
If you don't want to activate the card, just call the 1-800 number on the card, customer service and they will cancel the account with no hassle.

Please go ahead and inform that you do not need the card and want it canceled. However if it does not come with an annual fee you can keep the card because it helps in maintaining your credit. Its of course your personal choice to keep or return the card.

As far as credit score is concerned it will not take any drastic dip. A minor change might happen. So talk to the company I am sure they will tell you the way out.

The Capital One card I have has no annual fee. Call and ask them to wave that and ask for an increase in spending limit. Getting a $1000 should cover a family emergency. You can build credit by using the card and paying the balance every month.

Do not do it!! As posted in a previous answer I had them do it to me. I never used it but they kept adding finance charges and late fees to a card I never applied for. Cut it up and do not apply