Why isn't perpetual motion possible, even though we are so technologically advanced? Why perpetual motion wouldn't be possible if we are so technological advanced?
It is just a thing that I was wondering for too long. I mean, we are able to create so powerful permanent magnets, like neodymium magnets which can store an massive quantity of energy for a long period of time, but we are not able to use that energy at all. Why?
 A: We can create lots of things that can store energy for long periods of time - magnets, capacitators, batteries...but we must do work to put that energy in before taking it out. It's like a bank account that couldn't be overdrawn. Perpetual motion would be like a bank account that magically had money appearing in it every day without anyone depositing money into it.
A: "Nature, to be commanded, must be obeyed".
If perpetual motion is impossible in principle, no technology, no matter how advanced, will make it possible.
By the way, perpetual is longer than any long period of time.
A: From your link:

Perpetual motion describes motion that continues indefinitely without any external source of energy. This is impossible in practice because of friction and other sources of energy loss. Furthermore, the term is often used in a stronger sense to describe a perpetual motion machine of the first kind, a "hypothetical machine which, once activated, would continue to function and produce work" indefinitely with no input of energy. There is a scientific consensus that perpetual motion is impossible, as it would violate the first or second law of thermodynamics

(Bold mine)
In your questions three concepts are confused.
As somebody else has observed, "perpetual" is not synonymous to  "a long time" .
Secondly perpetual even if defined as "long time" is possible in the cosmos. The planetary system is an example. The moon revolves and will revolve about the earth for a very long time, the earth looses rotational kinetic energy by gravitational interactions, which creates our tides and from which tides we could get useful energy with clever valve systems.

Gravitational coupling between the Moon and the bulge nearest the Moon acts as a torque on Earth's rotation, draining angular momentum and rotational kinetic energy from Earth's spin. In turn, angular momentum is added to the Moon's orbit, accelerating it, which lifts the Moon into a higher orbit with a longer period. As a result, the distance between Earth and Moon is increasing, and Earth's spin slowing down.

Tidal energy  has been stored in our system from the time of its creation and we can take advantage of it for useful energy.
The third concept is the one I have highlighted in the first quote. You are asking whether we can find a setup in a lab where energy stored can be regained for very long times. (I hope it is clear that gaining energy from a machine the "forever" synonym of perpetual violates the laws of thermodynamics);  framed like this, the answer is: we are doing it continuously at hydroelectric plants, even burning fossil fuels.
So you are asking for a new long term source of energy not tapped up to now. Well, we managed to tap nuclear energy last century, that was new. So I would say it might be possible if our scientific knowledge breaks new frontiers. Present technological knowledge rests on the known physics, and known physics answers this question negatively.
A: For creating perpetual motion, storing energy is not important.
It does not help that we can store energy in magnets. We can store lots of energy in batteries too - nothing new here.
So maybe you think about the force a magnet can have in attracting or repelling something for infinite time without getting weaker? But if you just sit on a chair - there is force keeping you to fall down, and it will stay as long as you sit there, not getting weaker. Yes, the chair is repelling you just as a magnet could, both do not use up energy to keep the force. Nothing new here too.
Both is keeping a state perpetual. We could say perpetual non-motion.
To implement perpetual motion, we obviously need motion. And the motion needs to continue forever - or at least as long as we watch, without using up energy, from batteries or anything else - that's the very point of perpetual motion.  
But motion is causing all kinds of effects that use up energy.
If something moving touches something non-moving, there is friction, using up energy.
Didn't work.
Ok, we do not touch anything, we make it hover on magnets.
There is air drag.
Didn't work.
Vacuum? Just less air drag.
Didn't work.
Id does not help at all to make it "better" - it needs to be perfect in not using energy. 
Perfect vacuum? Good idea!  But there is no perfect vacuum.
Didn't work.  
Ok, storing large amounts of energy inside the system would make it better, of course. But we learned that "better" does not help us. We do not need to have extra energy, because if we even start to use up energy - planing to use it for infinite time -  we can give up.
That's what I do.
A: Looking at it from a very different perspective than my other answer:
Advanced technology or storing energy does not help with perpetual motion just as they do not help with making time go backwards.  
It is not in the scope of technology to begin with. 
The difference is: it looks like it is in the scope of technology - even if you look closely.
That may be the core of the problem, actually.
(Of course, if we manage to make time go backwards, we should reevaluate the perpetual motion thing...)
