Topology is at least partially built into human intuition because it talks about invariants - general properties and classification independent of fine details - exactly what humans are best at!
There are many real-life examples, for instance, the hairy ball theorem is encountered every time you wonder how come we can't have a map of the earth without having two poles where latitude is undefined. Or when trying to comb your hair. People who make clothes know very well that the more holes you need (sleeves,...), the more additional seams you need after you cut the main piece of cloth. The minimal number of seams is basically the first betti number. Making pants is fundamentally different from making a sweater, because the number of holes is different. Origami (or wrapping presents) falls into the same category.
Knots are very important in everyday life, even if you are not a mountain climber. You subconsciously realize it is important if something is a knot or not. If you tie your shoelaces correctly, you didn't make a true knot, so you can pull a strand and it unties. However, if you mess it up, you made a knot (and lost a few minutes to undo it). The same goes for earphones. While it may not be critically important to know how to classify knots and how to define all the complicated invariants seen in knot theory, on some basic level, you do grasp the concept of something being equivalent and just arranged differently, compared to something being fundamentally different. In one case, there is only a continuous rearrangement of the rope that separates you from falling to your death, in the other case, cutting is the only way.
Another example is map coloring: even if you don't make maps, as a child you probably tried coloring something so that adjacent fields don't share the same color. You probably even noticed that if you made a closed squiggle with a single stroke, an alternating pattern of two colors is enough. But nevertheless, it took mathematicians quite a long time (and a computer) to finally prove that you need at most 4 colours. That's nothing else but topology.
Of course, there is a thin line between topology (at least "practical" topology in 2D and 3D) and geometry, so there is an entire world of everyday problems where you have at least a hint of both.
In material science, the examples are: magnetic skyrmions and related solitons, which will help fit more data on a hard drive; liquid crystals have defects that are governed by quite complex topological rules - whether you want to avoid defects or control them, you need to know the rules; DNA knotting and topological insulators were already mentioned by other answers; topology of neural networks is a way of making sense of the mess of data you acquire in brain research; other topology-related physics questions are less "applicative" and could be regarded as purely academic: topology of curved space-time, study of knotted vortices in liquids, helicity of magnetic fields...
It depends on what you mean by "using" topology. In everyday life, you are dealing with some rudimentary topological notions subconsciusly, without actually performing any real math. In physical sciences, topology is currently sort of a hype: it has seen a high increase in research and publication volume - it will definitely yield new useful stuff, but there is also a lot of papers that are just there because it's interesting to look at something from topological perspective.
Both use cases actually use a very small subset of what mathematicians call topology. So for them, topology encountered in physics on the academic level isn't much better than counting poles on a globe.