Firstly, the paper doesn't sound very exciting. They are basically saying they have a large membrane that has some quantum related dynamics and the microbe is small enough that gluing it to the large thing won't change its motion enough to make it not get the quantum dynamics. And then on top they found a thing in the microbe sufficiently isolated from the rest of the microbe that it can be entangled with other things.
But you do have a serious misconception if you thought quantum mechanics is reserved for atoms. So you might benefit from a less mysterious description of quantum mechanics. For experts that are interested, I'll be describing the MIW version of quantum mechanics (which is not a misspelling of MWI).
First let's tall about classical mechanics. You can imagine a mass on a spring. If you plot velocity on the y axis and position on the x axis. Then you notice that you cab specify the position and the velocity (the initial conditions) jointly by specifying a point on the plane. And that over time the position and the velocity both change so you really get different points in the plane at different times. It always has to move clockwise and for the spring it actually moves on an ellipse. This is different than the earth moving on an ellipse around the sun because the y axis is velocity.
Next step, still classical mechanics. This time you have two particles. In different places. Each on a spring. So you can imagine a 4d space, two of them specifying where each particle is and the other two specifying the velocity of each particle. So now you have a point in 4d space moving around and its initial position is all the initial conditions of each particle and it just moves around in 4d always specifying the location and velocity of evey particle.
So that's what we expect from classical mechanics. You know the specification is in some range of your 4d space and you know how each point in the 4d space moves, so you know what happens. In general particles can move in 3d and there can be n particles so you need a point in 6n dimensional space to specify the system 3n for the location of each and 3n more for the velocity of each. So a single point in 6n dimensional space tells you everything classically and classical mechanics tells you how that point in 6n dimensional space moves in time.
In quantum mechanics you have a wavefunction. The wavefunction does some weird things. Firstly, it assigns velocities (from the probability current) for each location. So when you specify a point in 3n to describe a possible location where all the particles could be found, then the wavefunction specifies all 3n velocity components.
Let's visualize that. Go back to the single particle on a spring. It could be sitting at rest with the spring at its natural length. That's the being at the origin of the 2d space. Or it could start out displaced 1mm and at rest, that means starting it on the x axis and then going in a clockwise direction around an ellipse. If it was pulled out 2mm then it just goes on a larger ellipse. Once on an ellipse, it stats on that ellipse forever. But if you pulled it out 2mm and gave it a velocity then you moved it in the x and y direction in the 2d space effectively placing it on a larger ellipse and eventually it will be at larger than 2mm.
In quantum mechanics once you specify the wavefunction there is a velocity determined for each configuration of the particles. So back to the 2d picture. Classically, it could have any starting point on the 2d plane and then move along an ellipse. Now it is more like a bar graph. You see that each location along the x axis has to have a particular velocity.
That doesn't seem weird at first. But now imagine lots of points. If each one was a single classical world it would move along an ellipse. But because of this rule that each location have its one velocity consider two points on the same ellipse, to be concrete have one start on the positive y axis and the other one just a bit clockwise along the same ellipse. They can't just move along the ellipse like they classically would because if they did and that right most one got to the x axis it has to keep going clockwise but the places to the left already have their own velocities, positive ones. Its like you have two classical worlds each might want to move along the classical trajectory (in this case, the ellipse) but they can't, they get in others way. This means each point follows a different trajectory. That different trajectory is a purely quantum effect, and it always happens, no matter how many particles there are, how big something is, how massive, whatever. When it has more parts it just means you 2d space becomes 4d or 6000d (for 1000 particles moving in 3d).
When the space is larger it simply becomes easier for these two configurations to evolve so they don't get in each others way because there are so many directions to go.
So you can imagine it like a fluid in this large 6000d space, a fluid with some dye. You can think of it as a wave that is high only in a few places and the dye is a small tracer on the top of the water. The dye looks like a configuration and you can mathematically see it trace out its path. In the case of the spring different dye tracers track different configurations. If you have a hill, as the dye closer to the hill slows down the ones behind are forced to slow too and that slows the ones behind them. The net effect is that some dye tracers are seen to bounce away from the hill long before they get to the hill and some that classically would have slowed down and bounced back from the hill get pushed on based on the build up from behind.
We call that tunneling and reflection. And it is a quantum effect. But it isn't mysterious and it isn't reserved for small things. Sometimes the fluid separates part got pushed through, part for reflected and then if you are in a 6000d space with lots of things that can deflect you around those two bunches of fluid basically act like they are the only one.
These collections that are confined to a small region of the huge space but act like they are the only thing, that's the classical experience you know the configuration is something in a particular regions but you don't know exactly where and besides you are getting split every so often anyway.
When someone talks about quantum weirdness they are saying that two waves that could have acted on their own are set to overlap and thus they will make the tracers move in a nonclassical way because of that single velocity per location rule.
And the way you assign velocities to locations depends on the wavefunction, so lots of different wavefunctions can give lots of different dynamics.
Since you are the whole collection (not the/a tracer) you don't know coming into a splitting event that you will be going one way or the other. Both are going to happen. But each outcome will afterwards think of itself as the whole wave, simply because you won't ever meet the other wave.
So let's review what quantum mechanics is. There a huge space of configurations like 3000d each has a velocity (so a 3000d space of velocities) determined by the wavefunction. There are all regions with more tracers (or more fluid). So in this giant 6000d space there are regions with some fluid and regions without. Each little piece of fluid could act like an island and not notice the other islands as they move around it their own path through a huge 6000d space. That's is your classical experience. Sometimes a region breaks into two or more distinct regions as it moves around, that is what you experience as indeterminism. Each piece afterwards thinks of itself as the whole world and doesn't know why it went the way it did instead of the other way. Because you only experience which group you are in.
But each little tracer just follows its path. It doesn't always follow the classical path because it could spread out from the other tracers in the fluid and the fluid itself spreading out. But when the tracers are pushed in paths that deviate from the classical path they still do it in ways that conserve a kind of joint energy and momentum.
Now if you want to process information one way is to do it at the level of the whole region. So you specify the wavefunction which tells you the velocity of every tracer and the whole fluid and you track information from that. And some of that information can dynamically changes in a very classical way.
For instance in a hydrogen atom in the ground state the fluid pushing on the tracers is like a force that totally opposes the classical force and the tracer can sit there at rest, specifically the location of the electron and proton has a relative separation and that doesn't change in time. Quantum mechanics made that degree of freedom become dynamically boring. And the center of mass can just cruise along with no classical forces.
So one possible wavefunction has a the x for the proton and the x for the proton cruise along at the same speed (and same for the us and the zs) and you have different tracers for lots of differences between the $(x,y,z)$ of the electron and the $(x,y,z)$ of the proton. And the density of fluid and the tracers is (at each point) exactly what is needed to counter the classical electrostatic force they would exert on each other so they don't push each other away or towards each other.
So as a unit they act like a pair of particles with a fixed separation moving with a common velocity. So that center of mass moves in a very classical way. Similarly, there are regions of the fluid with information that can behave very classically. And that is what you are sensitive to.
So, there is no reason to expect some cutoff based on size. Size makes it easier for split regions to stay split. But it is just about making it easier.