I sold an item on craigslist, money was paid for it, but it was not picked up yet. I no longer wish to sell the item and I'm offering the person a full refund. The person is stating they don't want their money, just the items and that since they paid the items are technically theirs. Am I obligated to let them get the items? Or can I give them their money back?

How much time are we talking... days, weeks, months? Even years?

Was the original agreement that THEY come and pick them up?

She made the purchase last night and around 7 and I contacted her around 6:30 this morning.
I told her I wiould giver her a full refund but she is refusing, stating I must sell her the items or pay for her to get another set. She also, threatened me, so now I really want no dealings with her.
If she is refusing a refund, what do I do? Can I mail it certified mail?

The most she could make you do is refund what she actually paid, not what a new item costs... outside of let her have the items.

If you send them certified mail... it will be on your dime, and I'd get every kind of receipt you could, including making her show ID to the postman to get it... to cover your butt. SO there is no doubt WHO got them. Or have someone else act as an intermediary for you when she comes to get them, like a close friend or neighbor.

EDIT... don't do this alone... get some friends there when she intends to get it... just to have witnesses so nothing happens.

Will disagree smoothy, actually she can sue for the money paid but if this was a very good deal, she could sue for her loss if she has to pay more for another item. It is not done often but it has been done.

But you should give them the item, it is very poor business to list it, and then take the money but refuse to give the item.

There was a contract between the parties, an obvious understanding because the OP cashed the check (or wouldn't be talking about a refund). How far will purchaser go? I don't know.

If I somehow got into this situation I'd return the money with guaranteed funds, return receipt requested, and then hold my breath to see what happens next. Why the item was sold and then "unsold" will be important if the buyer takes the matter to Small Claims Court because she had to pay more for an identical item, she had a second item she could have purchased but didn't because she bought this item and now the second item is gone -

Small Claims Court is often about the principle (as opposed to the money) and I wouldn't be surprised if the buyer does file in SCC.