If you forced yourself to outfit a new kitchen solely with previously owned or inherited utensils and gear, you'd... actually end up with a decent toolset.
Not much changes in cooking day to day. The best tool for whisking is still a whisk. Steel pans will never go out of style. And the nature of cooking, with its heat and repetition and safety considerations, means every tool needs to be resilient. Most last forever and there are tons of great options for buying used, or working with hand-me-downs.
On the other hand, a few kitchen items should never be purchased used. Here they are, along with some alternatives....
PTFE or "Teflon" pans are produced using non-stick coatings that are famously prone to flaking off in tiny pieces—tiny pieces that end up in you and your family's food. Consuming enough of this material can cause flu-like symptoms, and the long term effects still aren't fully understood.
With prices on non-stick pans at an all-time low, there's simply no reason to risk your health on pans with obvious flaking, or even minuscule flaking that may escape your notice. Plus, new pans perform better.
Refrigerators & gas cooktops
Some large appliances tend to last longer or keep working without repair. On the other end of the spectrum are refrigerators and gas ranges.
According to HomeAdvisor, refrigerators are among the most often-repaired home appliances, with relatively frequent problems like compressor failures, rattling sounds, and icemaker or defroster issues. The average cost of a refrigerator repair in America is $200–400, but refrigerators are generally less expensive to repair if they're less than 5 years old.
Gas stoves—and all gas appliances, really—will always be more expensive to repair than their electric counterparts, due to risks like carbon monoxide leaks, or fires. It's usually not worth the trouble—nor the extra repair expenses—to buy these appliances used.
Soft or porous items
Used cloth items, like drying mats or kitchen towels, will need to be sanitized with high heat before use. Items like these, when damp, can become a breeding ground for mildew or worse over time. If you don't have access to a sanitizing washing machine (a rare feature), or you don't think the item will stand up to these temperatures, it isn't worth it.
Like cloth items, porous kitchen tools like wooden cutting boards or spoons can also house germs. A high heat dishwasher will sanitize a wooden cutting board, but you might break or bow your board while you're at it.
Things that dull
Be careful with honing steels and certain sharpening stones.
Most "dull" honing steels simply need a good cleaning. If that's the case, go ahead and buy used. But if we're talking about a decades-old steel that really has gone dull, there's no point in owning it.
Sharpening stones, a.k.a. water stones, are great for when you really get serious about blade care. But while fine stones—say 1000 grit—can finish off hundreds of knives, coarse stones don't last as long. A 120 grit stone will start to "pit" (become dull in certain spots) and require replacement after 30 or 40 sharpenings. Again, these are so cheap today, there's little value in buying used.
We don't often think of blenders as "things that dull," but blenders have blades too and they're much more difficult to sharpen.
Every blender's performance gets worse the longer it's used without replacing the blade. So, unless you're getting an amazing deal, consider a new blender instead. It's almost guaranteed to work better.
It wasn't until 1997 that scientists began to question the safety of bisphenol A or "BPA" in plastics used to store food. Today we still don't fully understand the effects, but the theories say this synthetic compound can leach into foods and cause everything from thyroid problems to neurological effects to cancer.
For our money, there's no point in taking the risk. Just about every new plastic food container or utensil advertises itself as BPA-free, and you can't put a price on peace of mind.