The easiest way to answer this question is to ask yourself, “How long can I wait for a replacement string when one breaks?”
A reality of playing the harp is that strings will break, and there’s really nothing you can do to prevent or anticipate it. Sometimes they break while you’re tuning, sometimes they go bang in the middle of the night, sometimes you lose one while you’re playing. Most of us would give anything for a magic device that could tell us which string was going to go next, but although there are reasons why strings break, there is no rhyme or pattern to it, at least not that we’ve ever found!
Another reality is that it’s not always quick and easy to find a replacement string wherever you are. Most lever harps are not like guitars, or even pedal harps, where a standard set of strings will work on pretty much any instrument. Each lever harp is designed with a specific set of strings in mind, and they are often custom-made for that particular model. If you replace a string with the wrong diameter or if the string wrap ends in the wrong place, the string may not sound very good, and at the very least you will have intonation and tone problems with the sharping lever, since each lever is sized very specifically to match its string. At the worst, if you replace a bunch of strings with the wrong ones, you could risk damaging your harp by putting more tension on it than it was designed to handle.
Because of this specificity, and because harp-focused music stores are not as thick on the ground as stores that carry guitars, the string you need may not be immediately available. If you’re lucky enough to live near a store that carries your make of harp, you might be able to walk in and pick up the string you need right away, but most people have to order their replacement harp strings online or over the phone and wait for the mail to deliver them.
One final thing to consider is that, as surprising as it seems, strings can’t be sent through the mail for the cost of a stamp. Even a single string in an envelope is too bulky and uneven to be processed by the postal service’s automated machines, so they return it to the sender and require a padded envelope and the postage rate for a small package. Because of this, shipping costs add up when you’re ordering strings one at a time. It’s a lot more economical in the long run to keep a full set on hand, use strings from the set as you need them, and then order a batch of strings all at once to refill your set.
So back to the original question – do you really need an extra set of strings? If you just play for your own enjoyment or in low-pressure situations and you don’t mind waiting a few days or a week for a replacement string, then you probably don’t need to have an extra set on hand. If, however, you do any performing or gigging, whether professionally or just volunteering at a church or nursing home (or if you’re horribly embarrassed to be seen with a gap-toothed harp), it’s definitely a good idea to have a backup set. When you’re facing an overnight shipping charge of $65 to get a replacement string in time for your wedding gig the following day, the cost of a full string set doesn’t seem so bad! Then again, maybe you can just alter your arrangements a little bit to squeak by without that one string…