As Rob pointed out, people use clip art in all sorts of areas. However, that can make things sort of difficult when figuring out how large to make them. Rule of thumb is to make things BIGGER vs. SMALLER: it is better to resize down than forcibly resize up (especially in web art). But ultimately, the size really depends on the design and what you're intending it to be used for.
Clip art for web use generally does not have to be gigantic in pixels or DPI, but again, it really depends on what kind of clip art you're making. A lot of people make social media icons or other navigational button sets. These icons usually have a pretty standard size of 256x256 down to 32x32. There are web standard banner sizes, too. Check out other web clip art sites and you'll definitely see standard dimensions pop out. ;)
Clip art for scrapbooking or things that are meant to be printed are another ball game. A resolution of 300 DPI is baseline because it's going to be printed. At that resolution, your images get huge fast. Once again, though, it depends on what the design is. Keep in mind that most people would be printing your clip art on A4 size paper at home. Take a regular piece of paper and think about how large you want whatever it is you're designing to be on that piece of paper. With that in mind, it's easier to create a blank canvas in Photoshop (or whatever program you use) with those approximate dimensions from the start and go from there.
Here on Etsy, both printed scrapbooking clip art pieces and web-use ones are very popular. They are very different markets and items, though, so do some research on other Etsy shops that have them and see what they sell. Also check out non-Etsy stock websites that cater to graphic designers. You'll see a lot of common info, I am sure. :)
