Bone conduction headphones are ideal for activities where it is vital to your safety to hear your surroundings. That's because they use a technology that sends audio vibrations through to the inner ear while leaving your ear canals open. Shokz is one of, if not the best, brand making bone conduction headphones right now; at the high end of Shokz's offerings is the OpenRun Pro, which are currently discounted on Amazon for $139.94 (originally $179.95).
If you've never used bone conduction headphones, they're worth a try. They've come a long way since their early days. The major downside of the way they work has been that the bass doesn't sound as good as their in-ear counterparts. The OpenRun Pro is designed to compensate for some of that—you won't get the same thumpy bass you'd get from regular headphones, but it'll be the best you can get from any other bone conduction headphones.
As the name implies, these are meant to be used on runs outdoors. They have an IP55 rating for dust and water resistance, are lightweight and comfortable, and, because of their design, can be worn with bicycle helmets and sunglasses.
You can use these as everyday headphones around the house as well since they can take calls; just be mindful that they don't block any sound. If you plan to use these at a gym that's playing music or has a lot of ambient noise, you'll hear everything, so you might prefer taking regular headphones or wearing earplugs (but that will defeat the purpose of the "open ear" headphones).